Schwartz Analysis and IntertwiningDistributionsbyXinyu LiuB.Sc., University of Science and Technology of China, 2007M.Sc., University of Science and Technology of China, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2017c© Xinyu Liu 2017AbstractIn this dissertation, we combine the work of A. Aizenbud and D. Goure-vitch on Schwartz functions on Nash manifolds, and the work of F. du Clouxon Schwartz inductions, to develop a toolbox of Schwartz analysis. We thenuse these tools to study the intertwining operators between parabolic induc-tions, and study the behavior of intertwining distributions on certain opensubsets. Finally we use our results to give new proof of results in [15] onirreducibilities of degenerate principal series and minimal principal series.iiLay SummaryParabolic inductions and intertwining operators between them are ofgreat importance in representation theory. The structure of parabolic in-ductions is still mysterious, and only sporadic results have been obtained.Our final goal is to find a unified approach to study intertwining op-erators between parabolic inductions, by using only the structures of thequotient spaces and the information about the representations to be in-duced.This dissertation shows the very first step of the entire framework. Wecombine the work of A. Aizenbud, D. Gourevitch and F. du Cloux, developan algebraic toolbox, and use the results to study the irreducibility of unitaryparabolic inductions. We give a new proof of results on irreducibilities ofdegenerate principal series and minimal principal series in [15].iiiPrefaceThis dissertation is the original, unpublished work of the author, XinyuLiu.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 A Brief Summary of the Thesis . . . . . . . . . . . . . . . . . 11.1.1 Goal of the Thesis . . . . . . . . . . . . . . . . . . . . 11.1.2 Distribution Analysis (Chapter 5) . . . . . . . . . . . 21.1.3 Organization of the Thesis . . . . . . . . . . . . . . . 51.1.4 Main Body of the Thesis (Chapter 6—9) . . . . . . . 81.2 History and Motivation . . . . . . . . . . . . . . . . . . . . . 121.2.1 The Work of Bruhat [15] . . . . . . . . . . . . . . . . 131.2.2 Some Other Related Works . . . . . . . . . . . . . . . 171.3 Reading Guide and Key Points of the Chapter 2, 3 and 4 . . 201.3.1 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . 211.3.2 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . 211.3.3 Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . 221.4 Convention on Notation . . . . . . . . . . . . . . . . . . . . . 242 Basic Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.1 Groups Studied in the Thesis . . . . . . . . . . . . . . . . . . 282.1.1 Algebraic Groups in this Thesis . . . . . . . . . . . . 292.1.2 G as a Lie Group . . . . . . . . . . . . . . . . . . . . 322.2 NF-Spaces and DNF-Spaces . . . . . . . . . . . . . . . . . . 35v2.2.1 Basic Notions on Topological Vector Spaces . . . . . 352.2.2 Properties of NF-Spaces and DNF-Spaces . . . . . . . 362.3 Representations Studied in the Thesis . . . . . . . . . . . . . 392.3.1 Basic Notions on Representations . . . . . . . . . . . 392.3.2 Harish-Chandra Modules . . . . . . . . . . . . . . . . 422.3.3 Harish-Chandra Representations . . . . . . . . . . . . 432.3.4 Smooth Inductions . . . . . . . . . . . . . . . . . . . 432.3.5 Frobenius Reciprocity . . . . . . . . . . . . . . . . . . 452.3.6 (Hilbert) Normalized Parabolic Inductions . . . . . . 453 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.1 Real Algebraic Varieties . . . . . . . . . . . . . . . . . . . . . 483.1.1 Algebraic Subsets of Rn . . . . . . . . . . . . . . . . . 483.1.2 Affine Real Algebraic Variety . . . . . . . . . . . . . . 493.2 Basic Notions on Nash Manifolds . . . . . . . . . . . . . . . 503.2.1 Semi-algebraic Sets and Maps . . . . . . . . . . . . . 503.2.2 Nash Submanifolds of Rn . . . . . . . . . . . . . . . . 513.2.3 Restricted Topology, Sheaf and Cosheaf . . . . . . . . 523.2.4 Affine Nash Manifolds and Abstract Nash Manifolds . 543.3 Bruhat Decompositions . . . . . . . . . . . . . . . . . . . . . 553.3.1 Some Terms on Group Actions . . . . . . . . . . . . . 553.3.2 Bruhat Decomposition . . . . . . . . . . . . . . . . . 573.3.3 Closure Order on Double Cosets . . . . . . . . . . . . 593.4 Algebraic Preliminary . . . . . . . . . . . . . . . . . . . . . . 623.4.1 Left Torsion Submodule . . . . . . . . . . . . . . . . . 633.4.2 Right Torsion Submodule . . . . . . . . . . . . . . . . 653.4.3 Tensor Product Module . . . . . . . . . . . . . . . . . 663.4.4 Torsion Submodule of Tensor Products—I . . . . . . 673.4.5 Depth and Torsion Modules . . . . . . . . . . . . . . 693.4.6 Finite Dimensional Torsion Modules . . . . . . . . . . 713.4.7 Torsion Submodule of Tensor Products—II . . . . . . 734 Schwartz Functions, Schwartz Inductions and Schwartz Dis-tributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.1 Some Notions . . . . . . . . . . . . . . . . . . . . . . . . . . 804.1.1 Nash Differential Operators . . . . . . . . . . . . . . 804.1.2 Affine Real Algebraic Varieties as Affine Nash Mani-folds . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.2 Schwartz Functions on Affine Nash Manifolds . . . . . . . . . 824.2.1 Definition of Schwartz Functions . . . . . . . . . . . . 82vi4.2.2 Properties of Schwartz Functions . . . . . . . . . . . 834.2.3 Complex-Valued Schwartz Functions . . . . . . . . . 854.3 Vector Valued Schwartz Functions . . . . . . . . . . . . . . . 874.3.1 Vector Valued Schwartz Functions on Affine Nash Man-ifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.3.2 Summary of Properties of Schwartz Functions . . . . 904.4 Schwartz Distributions . . . . . . . . . . . . . . . . . . . . . 914.4.1 Sheaf of Schwartz Distributions . . . . . . . . . . . . 924.4.2 Extension of Schwartz Distributions from Closed NashSubmanifolds . . . . . . . . . . . . . . . . . . . . . . . 934.4.3 Distributions Supported in Closed Subsets . . . . . . 944.5 Application to Nonsingular Affine Real Algebraic Varieties . 964.5.1 Pseudo-Sheaf and Pseudo-Cosheaf . . . . . . . . . . . 964.5.2 Schwartz Functions and Distributions on NonsingularAffine Real Algebraic Varieties . . . . . . . . . . . . . 994.5.3 Supports and Maximal Vanishing Subsets of Distribu-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.5.4 Geometry on G = G(R) . . . . . . . . . . . . . . . . 1014.5.5 Right Regular Actions on Schwartz Function Spaces . 1034.6 Schwartz Inductions . . . . . . . . . . . . . . . . . . . . . . . 1054.6.1 Schwartz Induction SIndGPσ . . . . . . . . . . . . . . 1054.6.2 Local Schwartz Inductions . . . . . . . . . . . . . . . 1084.6.3 Open Extensions and Closed Restrictions of SchwartzInductions . . . . . . . . . . . . . . . . . . . . . . . . 1094.6.4 Pseudo-Cosheaf Property of Schwartz Inductions . . . 1124.6.5 Distributions on Schwartz Induction Spaces . . . . . 1144.6.6 Group Actions on Local Schwartz Inductions . . . . . 1164.6.7 Local Schwartz Inductions on Fibrations . . . . . . . 1185 Intertwining Distributions . . . . . . . . . . . . . . . . . . . . 1235.1 Intertwining Distributions . . . . . . . . . . . . . . . . . . . . 1255.1.1 Intertwining Distributions . . . . . . . . . . . . . . . 1255.2 Supports of Intertwining Distributions . . . . . . . . . . . . . 1275.2.1 Some Topological Facts . . . . . . . . . . . . . . . . . 1275.2.2 The suppD and vanD are Right Q-Stable . . . . . . . 1285.2.3 The suppD and vanD are Left P -Stable . . . . . . . 1295.3 Some Notions of Distribution Analysis on G . . . . . . . . . 1325.3.1 Maximal Double Cosets in Supports . . . . . . . . . . 1325.3.2 Diagonal Actions on Distribution Spaces—I . . . . . 1335.3.3 Diagonal Actions on Distribution Spaces—II . . . . . 135vii5.4 Self-Intertwining Distributions . . . . . . . . . . . . . . . . . 1365.4.1 (P, PΩ)-Stable Subsets of G and Local Schwartz In-ductions . . . . . . . . . . . . . . . . . . . . . . . . . 1375.4.2 Scalar Intertwining Operators . . . . . . . . . . . . . 1395.4.3 Analysis of Schwartz Distributions . . . . . . . . . . . 1395.5 The Irreducibility of Unitary Parabolic Inductions . . . . . . 1415.5.1 General Case suppD * P . . . . . . . . . . . . . . . . 1435.5.2 Extreme Case suppD ⊂ P . . . . . . . . . . . . . . . 1445.5.3 Irreducibility of I . . . . . . . . . . . . . . . . . . . . 1456 Schwartz Distributions Supported in Double Cosets . . . . 1476.1 Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1496.1.1 Formulating the Main Theorem . . . . . . . . . . . . 1506.1.2 Gw and its Tubular Neighbourhood Zw . . . . . . . . 1526.1.3 Change of Neighbourhoods . . . . . . . . . . . . . . . 1536.2 Schwartz Distributions with Point Supports . . . . . . . . . . 1556.3 Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . 1566.3.1 The Kernel Ker(resw) . . . . . . . . . . . . . . . . . . 1576.3.2 The Isomorphism (6.9) . . . . . . . . . . . . . . . . . 1596.3.3 The Kernel Ker(Resw) and the Isomorphism (6.10) . 1606.3.4 General Case of F . . . . . . . . . . . . . . . . . . . . 1617 Torsions of Distributions . . . . . . . . . . . . . . . . . . . . . 1627.1 Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1637.1.1 Main Theorem in this Chapter . . . . . . . . . . . . . 1647.1.2 Left vs. Right . . . . . . . . . . . . . . . . . . . . . . 1667.1.3 Setting and Notation for the Proof . . . . . . . . . . 1697.1.4 A Formula . . . . . . . . . . . . . . . . . . . . . . . . 1707.1.5 Geometric Preparation . . . . . . . . . . . . . . . . . 1747.2 The Inclusion M⊂ K[n∅•] . . . . . . . . . . . . . . . . . . . . 1817.2.1 Some Algebraic Lemmas—I . . . . . . . . . . . . . . 1817.2.2 The Inclusion Mk ⊂ K[n∅•] . . . . . . . . . . . . . . . 1847.3 The Inclusion K[n∅•] ⊂M . . . . . . . . . . . . . . . . . . . . 1867.3.1 Orders on Multi-index Sets . . . . . . . . . . . . . . . 1887.3.2 Some Algebraic Lemmas—II . . . . . . . . . . . . . . 1927.3.3 Heights on Basis of n−w . . . . . . . . . . . . . . . . . 1957.3.4 The Coefficients Ci . . . . . . . . . . . . . . . . . . . 1967.3.5 The Elements Yk · Y I . . . . . . . . . . . . . . . . . . 1977.3.6 The Submodule KI . . . . . . . . . . . . . . . . . . . 1997.3.7 The Inclusion K[n∅•] ⊂M . . . . . . . . . . . . . . . . 203viii8 Shapiro’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 2058.1 Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2068.1.1 Main Object of This Chapter . . . . . . . . . . . . . . 2068.1.2 The Annihilator (I ′w)[n∅k] and the Annihilator-InvariantTrick . . . . . . . . . . . . . . . . . . . . . . . . . . . 2078.2 The Iw and The Tensor Product Iw ⊗ Fk . . . . . . . . . . . 2108.2.1 Revisiting the Iw . . . . . . . . . . . . . . . . . . . . 2118.2.2 The Iw is Isomorphic to SIndN∅N∅∩w−1Pwσw . . . . . . 2138.2.3 The Isomorphism (8.5) is M∅-Equivariant . . . . . . . 2168.2.4 External Tensor Products . . . . . . . . . . . . . . . . 2178.2.5 Tensor Product of Schwartz Inductions . . . . . . . . 2188.3 Shapiro’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . 2238.3.1 Shapiro’s Lemma . . . . . . . . . . . . . . . . . . . . 2248.3.2 Applying Shapiro’s Lemma . . . . . . . . . . . . . . . 2269 Application: Finite Dimensional (σ, V ) . . . . . . . . . . . . 2299.1 Degenerate Principal Series . . . . . . . . . . . . . . . . . . . 2299.1.1 A∅-Spectrum on (τ, V ) . . . . . . . . . . . . . . . . . 2309.1.2 Formulating an Analogue of Bruhat’s Theorem 7.4 . . 2339.1.3 The Space HomP (V′, (I/I>e)′) . . . . . . . . . . . . . 2349.2 The Space HomP∅(V′, (I≥w/I>w)′) . . . . . . . . . . . . . . . 2369.2.1 The Reason to Study n∅-Torsion Subspaces . . . . . . 2369.2.2 Comparison of the A∅-Spectrums . . . . . . . . . . . 2379.3 Minimal Principal Series . . . . . . . . . . . . . . . . . . . . 2409.3.1 An Analogue of Bruhat’s Theorem 7.2a . . . . . . . . 2409.3.2 Real Parts of the A∅-Spectrum . . . . . . . . . . . . . 24210 General Results and Future Work . . . . . . . . . . . . . . . 24510.1 Generalization of Chapter 6 . . . . . . . . . . . . . . . . . . 24510.1.1 Formulating the Theorem . . . . . . . . . . . . . . . . 24510.1.2 Sketch of the Proof . . . . . . . . . . . . . . . . . . . 24610.2 Generalization of Chapter 7 . . . . . . . . . . . . . . . . . . 24810.2.1 Formulating the Theorem . . . . . . . . . . . . . . . . 24810.2.2 Sketch of the Proof . . . . . . . . . . . . . . . . . . . 24910.3 Generalization of Chapter 8 . . . . . . . . . . . . . . . . . . 25210.4 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . 25310.4.1 A Conjecture . . . . . . . . . . . . . . . . . . . . . . . 25410.4.2 Another Conjecture of Casselman . . . . . . . . . . . 255Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257ixList of Figures4.1 Independent of Neighbourhoods . . . . . . . . . . . . . . . . . 955.1 Schwartz and Distribution-Open . . . . . . . . . . . . . . . . 1405.2 Schwartz and Distribution-Closed . . . . . . . . . . . . . . . . 1416.1 Restriction maps and kernels . . . . . . . . . . . . . . . . . . 1487.1 Ker(resw) and Ker(Resw) as transverse derivatives . . . . . . 1657.2 Linear order and Reverse lexicographic order (d=2) . . . . . . 190xAcknowledgementsMy deep gratitude goes to my advisor Prof. Bill Casselman, for intro-ducing me to many beautiful areas of representations theory, especially thesubject studied in this dissertation. While working on this dissertation, alot of difficulties were overcome after discussion with him. The crucial ideato consider torsion subspaces also belongs to him. In a word, this thesiswould never exist without his encouragement and financial support in thelast six years.I would like to thank the members of my PhD committee: Prof. JuliaGordon and Prof. Lior Silberman, for creating a friendly and encouragingatmosphere.I also thank the secretaries in the main office, especially Lee Yupitun,Roseann Kinsey, Marlowe Dirkson, Ann Artuso and Jessica Trat. Theirhard work made my study and life in UBC much easier.Finally I want to thank my parents for their support and understandingof my choice to study mathematics for so many years.xiChapter 1Introduction.1.1 A Brief Summary of the Thesis1.1.1 Goal of the ThesisThe long-term goal of our work is to study intertwining operators be-tween two parabolic inductions, by using the tools of Schwartz analysisdeveloped in [1], [2], [20] and Chapter 4 of this thesis.In this thesis, to simplify the problem, we will work under the followingbasic setting:• The Lie groups studied in the thesis, are the real point groups of con-nected reductive linear algebraic groups defined over R.• We will only work on smooth inductions of unitary representations,namely for a unitary representation (τ, V ) of a parabolic subgroup P ,we study the smooth induction C∞IndGP (τ ⊗ δ1/2P ), which is infinitesi-mally equivalent to the ordinary (Hilbert) normalized induction IndGP τ .And temporarily we only study the irreducibilities of such representa-tions.Many sporadic results have been obtained for irreducibilities of parabolicinductions. Most of them follow Mackey’s machinery, and prove the irre-ducibility by finding upper bound of certain multiplicities. Some other peo-ple use pure algebraic tools especially for degenerate principal series. Ouraim is to find a unified approach to study intertwining operators, by onlyusing the geometric structure on the group (or flag manifolds) and resultsin pure representation theory. As the reader can see in the main body ofthis thesis (Chapter 5—9), we prove irreducibilities by geometric/algebraictricks, combined with some fundamental results in representation theory(e.g. Frobenius reciprocity, Schur lemma etc). The only unusual result wehave used is Shapiro’s Lemma, but we only use the special case on 0thcohomology, which is an extension of the Frobenius reciprocity.In this thesis, we first develop a toolbox of Schwartz analysis on algebraicgroups. Then we study intertwining operators as intertwining distributions,1classify them by their supports, and study their local behavior by restrictingthe distributions to open neighbourhoods of double cosets. In the local studyof intertwining distributions, we combine the tools of transverse derivatives,torsion subspaces and Shapiro’s Lemma. In the current version of thesis,we are able to prove the results analogous to Theorem 7.2 and 7.4 in [15],about irreducibilities of minimal principal series and degenerate principalseries, for the above class of Lie groups.In the following three subsections 1.1.2, 1.1.3 and 1.1.4, we give a sketchyoutline of the thesis.1.1.2 Distribution Analysis (Chapter 5)The central idea of our study is to realize intertwining operators (be-tween two parabolic inductions) as “intertwining distributions on Schwartzinduction spaces”, and study the local properties of such distributions. Thestrong confinement from the local properties will tell us when there is nonon-scalar intertwining distributions (operators), thus the unitary parabolicinductions are irreducible. This method also partially tells us what the in-tertwining distributions look like on certain open subsets. One object of ourfuture study is to answer how to the intertwining distributions on the entiregroup.Intertwining Distributions (5.1, 5.2, 5.3)Let G be a connected reductive linear algebraic group defined over R, letP,Q be two parabolic R-subgroups of G, let G,P,Q be the correspondingreal point groups. Let (σ1, V1) be a Harish-Chandra representation of P (see2.3), and (σ2, V2) be a Harish-Chandra representation of Q. Let C∞IndGPσ1(resp. C∞IndGQσ2) be the space of (unnormalized) smooth induction of σ1(resp. σ2) from P (resp. Q) to G. Since the P,Q are parabolic sub-groups, the quotient manifolds P\G,Q\G are compact, and the above twosmooth inductions equal to the corresponding Schwartz inductions (see 4.6for the term “Schwartz inductions”): C∞IndGPσ1 = SIndGPσ1, C∞IndGQσ2 =SIndGQσ2. For simplicity, we denote them by I = SIndGPσ1, J = SIndGQσ2,and let HomG(I, J) be the space of intertwining operators from I to J .To study the intertwining operators in HomG(I, J), we first apply Frobe-nius reciprocity, i.e. the isomorphism:HomG(I, J)∼−→ HomQ(I, V2), T 7→ Ωe ◦ T.Thus we identify the space HomG(I, J) of intertwining operators, with the2space HomQ(I, V2) ofQ-equivariant V2-valued distributions on I = SIndGPσ1.We call such distributions intertwining distributions. In this thesis, wewill work on the case when Q = P, σ2 = σ1.Self-Intertwining Distributions and Irreducibility (5.4)Let P = PΘ be a parabolic R-subgroup of G corresponding to a subsetΘ of the base of restricted roots, with real point group P = PΘ, let (σ, V ) bea Harish-Chandra representation of P , and let I = SIndGPσ (= C∞IndGPσ)be the Schwartz (smooth) induction space. We are interested in the spaceHomG(I, I) of self-intertwining operators on I.In particular, when σ = τ⊗δ1/2P where τ is a unitary representation on V ,the I is irreducible if and only if HomG(I, I) = C. Actually the I is infinites-imally equivalent to the normalized unitary (Hilbert) parabolic inductionIndGP τ (see 2.3.6). The I and IndGP τ are irreducible/reducible simultaneously.And they are irreducible if and only if HomG(I, I) = HomG(IndGP τ, IndGP τ) =C. The main body of the thesis (Chapter 6 to 9), are devoted to the study ofirreducibility of I. By the above discussion, we can show the irreducibilityof I (equivalently IndGP τ), by showing the space HomP (I, V ) of intertwiningdistributions is one dimensional.The Supports and Maximal Double Cosets (5.2, 5.3, 5.4)We consider self-intertwining distributions and keep the above settingon G,P = PΘ, τ, σ. Let D ∈ HomP (I, V ) be an intertwining distribution.We will show in 5.2 that every intertwining distribution has a well-definedsupport, denoted by suppD, and it is a (real) Zariski closed union of (P, P )-double cosets. Obviously, the intertwining distributions corresponding toscalar intertwining operators have their supports contained in P .Actually for any algebraic subgroup H of P , the suppD is also a unionof (P,H)-double cosets. In particular, for any subset Ω ⊂ Θ, let PΩ bethe corresponding standard parabolic R-subgroup with PΩ be its real pointgroup. Then the support suppD is also a union of (P, PΩ)-double cosets.There are finitely many (P, PΩ)-double cosets on G, parameterized by theset [WΘ\W/WΩ] of minimal representatives (see 3.3), and they are orderedby the closure order (see 3.3) by the real Zariski topology.For each Ω ⊂ Θ, and a representative w ∈ [WΘ\W/WΩ], we denote by3GΩw = PwPΩ the corresponding (P, PΩ)-double coset, and letGΩ≥w =∐PwPΩ⊂PxPΩPxPΩGΩ>w = GΩ≥w −GΩwThese two are Zariski open subsets of G, and GΩw is closed in GΩ≥w with opencomplement GΩ>w. For the (P, PΩ)-stable subsets GΩ≥w, GΩ>w, GΩw, we denotethe corresponding local Schwartz inductions (see 4.6.2) byIΩ≥w = SIndGΩ≥wP σ, IΩ>w = SIndGΩ>wP σ, IΩw = SIndGΩwP σ.They are smooth PΩ-representations under the right regular PΩ-actions. Wehave the inclusions (of PΩ-representations): IΩ>w ⊂ IΩ≥w ⊂ I for each Ω ⊂ Θand w ∈ [WΘ\W/WΩ].Since there are finitely many (P, PΩ)-double cosets on G, one can choosea (non-unique) maximal (P, PΩ)-double coset (under the closure order) con-tained in suppD, say GΩw. Then the restriction of the intertwining distribu-tion D ∈ HomP (I, V ) to the subspace IΩ≥w is non-zero, and the restrictionof D to the subspace IΩ>w is zero.Restricting the Intertwining Distributions (5.5)Let D ∈ HomP (I, V ) be an intertwining distribution with its supportdenoted by suppD. There are two special cases: (1) the suppD is containedin P (the identity double coset PeP ); (2) the suppD is not contained in P .We will study these two cases separately.Suppose suppD is contained in P . Let I>e = SIndG−PP σ be the localSchwartz induction of σ from P to the open complement G − P , then I>eis a P -subrepresentation of I, and D vanishes on this subspace. ThereforeD : I → V factor through the quotient space I/I>e, and let D be thecorresponding map D : I/I>e → V . Obviously D is still P -equivariant whenthe quotient is endowed with the quotient P -action. Its adjoint map D∗:V ′ → (I/I>e)′ is also a P -equivariant continuous linear map. Since V andI/I>e are nuclear hence reflexive, we have the following linear isomorphism:{D ∈ HomP (I, V ) : suppD ⊂ P} ∼−→ HomP (V ′, (I/I>e)′), D 7→ D∗.Suppose suppD is not contained in P . Then for each subset Ω ⊂ Θ,as above we can find a maximal (P, PΩ)-double coset contained in suppD,say GΩw for some w ∈ [WΘ\W/WΩ]. Since GΩw is maximal in the support,4the restricted distribution DΩ≥w : IΩ≥w → V is PΩ-equivariant, nonzero andvanishes on the subspace IΩ>w. Similar to the above discussion, it factorthrough the quotient IΩ≥w/IΩ>w, and the corresponding map is denoted byDΩ≥w : IΩ≥w/IΩ>w → V . This map is also PΩ-equivariant and non-zero. By thesame argument as above, the correspondence D 7→ (DΩ≥w)∗ gives a bijection:{D : GΩw is maximal in suppD} ∼−→ HomPΩ(V ′, (IΩ≥w/IΩ>w)′)− {0}Combining the above two cases, to show the irreducibility of I, we justneed to:• Show the HomP (V ′, (I/I>e)′) = C.• Find a subset Ω ⊂ Θ, and for all w ∈ [WΘ\W/WΩ] and w 6= e, showthe HomPΩ(V′, (IΩ≥w/IΩ>w)′) = {0}.1.1.3 Organization of the ThesisBy the above discussion, we are required to study the local behavior of in-tertwining distributions, namely the dual quotients (IΩ≥w/IΩ>w)′ (the (I/I>e)′is the special case when w = e), and the spaces HomPΩ(V′, (IΩ≥w/IΩ>w)′), forall Ω ⊂ Θ and w ∈ [WΘ\W/WΩ].General AttemptsWe have made progress in the following three steps:• Step 1: Express the dual quotients (IΩ≥w/IΩ>w)′ as transversederivatives. The main result in the first step is summarized as The-orem 10.1, namely, the (IΩ≥w/IΩ>w)′ is isomorphic to the space of dis-tributions on IΩ≥w supported in GΩ>w, and the latter space consistsof transverse derivatives of distributions on GΩw. More precisely thefollowing map is an isomorphism:(IΩw )′ ⊗ U(tΩw)→ (IΩ≥w/IΩ>w)′,where the tΩw is the transverse subalgebra nΩ ∩ w−1nPw.• Step 2: Study the torsion subspace on (IΩ≥w/IΩ>w)′ ' (IΩw )′ ⊗U(tΩw). To make the spaces HomPΩ(V′, (IΩ≥w/IΩ>w)′) computable, weneed the special properties on V ′. The following two special casescaught our attention:5(1) If (τ, V ) is P -irreducible and unitary, then the NP = NΘ actstrivially on V and V ′. If we let Ω = Θ, then an arbitrary mapΦ ∈ HomPΘ(V ′, (IΘ≥w/IΘ>w)′) has its image in the nΘ-invariantsubspace of (IΘ≥w/IΘ>w)′. Therefore we haveHomPΘ(V′, (IΘ≥w/IΘ>w)′) = HomMΘ(V′, [(IΘ≥w/IΘ>w)′][nΘ1]),where the [(IΘ≥w/IΘ>w)′][nΘ1] is the nΘ-invariant subspace. How-ever, computing the nΘ-invariant subspace on(IΘ≥w/IΘ>w)′ ' (IΘw )′ ⊗ U(tΘw)is infeasible since the tensor product is not a module tensor prod-uct. We do not have a good algebraic description of the nΘ-invariant subspace [(IΘ≥w/IΘ>w)′][nΘ1]. To compromise, we com-pute the nΘ-torsion subspace [(IΘ≥w/IΘ>w)′][nΘ•], and we have thefollowing inclusion:HomPΘ(V′, (IΘ≥w/IΘ>w)′) ↪→ HomMΘ(V ′, [(IΘ≥w/IΘ>w)′][nΘ•]).(2) If V is n∅-torsion (e.g. V is finite dimensional or certain discreteseries), then the n∅ acts nilpotently on V and V ′. In particular,the V ′ equals to its n∅-torsion subspace, and an arbitrary Φ ∈HomP∅(V′, (I≥w/I>w)′) has its image in the n∅-torsion subspaceof (I≥w/I>w)′. Therefore we haveHomP∅(V′, (I≥w/I>w)′) = HomM∅(V′, [(I≥w/I>w)′][n∅•]).The above two special cases urge us to compute the torsion subspaceson the dual quotients. To deal with all cases simultaneously, we con-sider the nΩ-torsion subspaces on (IΩ≥w/IΩ>w)′ ' (IΩw )′ ⊗ U(tΩw) for allΩ ⊂ Θ. The main result in the second step is summarized as Theorem10.3, namely we have:[(IΩw )′ ⊗ U(tΩw)][n•Ω] = [(IΩw )′][n•Ω] ⊗ U(tΩw).• Step 3: The nΩ-torsion subspace on (IΩw )′. The above two stepsrequire us to find the MΩ-action on the torsion subspace [(IΩw )′][n•Ω] ormore precisely on the annihilators [(IΩw )′][nkΩ] for all k ≥ 0. This stepis comprised by the following three steps:6(1) First we use the annihilator-invariant trick to show[(IΩw )′][nkΩ] = H0(nΩ, (IΩw ⊗ FΩk )′),where FΩk = U(nΩ)/(nkΩ) is a finite dimensional representation ofNΩ (or even PΩ).(2) Second we show the local Schwartz induction IΩw is isomorphic toanother Schwartz induction space:IΩw ' SIndPΩPΩ∩w−1Pwσwwhere σw = σ ◦Adw is the twisted representation of σ.(3) Third we use the tensor product trick to showIΩw ⊗ FΩk ' SIndPΩPΩ∩w−1Pw(σw ⊗ FΩw ).In sum, we have[(IΩw )′][nkΩ] ' H0(nΩ,SIndPΩPΩ∩w−1Pw(σw ⊗ FΩw )),and we just need to study the MΩ-action on it. Currently, we are onlyable to compute the above space for Ω = ∅ by using Shapiro’s lemma,and we will explain the reason below.The Reason to Consider the Case Ω = ∅In the above step 3, we meet the two main obstacles for further study:1. First we need to find the explicit MΩ-action on the tensor product[(IΩw )′][n•Ω] ⊗ U(tΩw).However, this is not a tensor product of MΩ-representations for someof the Ω (including Θ itself), as one can see the tΩw = nΩ ∩w−1nPw isnot stable under the MΩ-conjugation. This will make the MΩ-actionon [(IΩw )′][n•Ω] ⊗U(tΩw) very complicated. Fortunately, when Ω = ∅, thet∅w = n∅∩w−1nPw is M∅-stable and the tensor product (I ′w)[n∅•]⊗U(t∅w)is indeed a tensor product of M∅-representations.2. Second we are only able to compute the space of invariant distributionsH0(nΩ,SIndPΩPΩ∩w−1Pw(σw ⊗ FΩw ))7for the case Ω = ∅. This is because the quotient P∅∩w−1Pw\P∅ is N∅-transitive, and we can apply Shapiro’s lemma to compute the above0th invariant space. But in general the quotient PΩ ∩ w−1Pw\PΩ isnot NΩ-transitive, and Shapiro’s lemma does not apply to the generalcase.For the above two reasons, we are only able to obtain results of irre-ducibilities, by applying the above three steps, to the case when Ω is theempty set. This case already includes the interesting cases of degenerateprincipal series and minimal principal series.To make the thesis clear and neat, in the main body of the thesis (Chapter6—8, we choose Ω to be the empty set, show the results in the above threesteps, and in Chapter 9 we apply them to reproduce the Theorem 7.2a and7.4 in [15]. We put the general case of Ω in the Chapter 10, since we arecurrently unable to apply our techniques to obtain useful results.1.1.4 Main Body of the Thesis (Chapter 6—9)Currently, we are only able to prove results on irreducibility, by using ourgeneral results in the special case when Ω is the empty set. In the main bodyof the thesis (Chapter 6—9), we will only consider the case Ω = ∅. We simplydrop all superscript Ω, and use the notations G≥w, G>w, Gw, I≥w, I>w, Iw.We will study the dual-quotients (I≥w/I>w)′ for all w ∈ [WΘ\W ], and showthe irreducibility of I by showing:HomP (V′, (I/I>e)′) = CHomP∅(V′, (I≥w/I>w)′) = {0}, ∀w 6= e, w ∈ [WΘ\W ]under certain conditions on σ (or τ). The three general steps shown in thelast subsection correspond to the chapters 6, 7 and 8. And in the Chapter 9,we use the results in the three steps (three chapters) to reproduce analogousresults of Theorem 7.2a and 7.4 in [15].Transverse Derivatives (Chapter 6)The first step is to write elements in the dual-quotient (I≥w/I>w)′ astransverse derivatives. The space (I≥w/I>w)′ is exactly the kernel of thefollowing restriction map of scalar distributions:Resw : I′≥w → I ′>w,i.e. the elements in (I≥w/I>w)′ are exactly the scalar distributions on I≥wthat vanish on the subspace I>w. Similar to the classical results on Euclidean8spaces, we will show such distributions are given by transverse derivativesof distributions on Gw (see 6.1.1 for detailed explanation):Theorem (Theorem 6.1). The natural mapI ′w ⊗ U(n−w)→ Ker(Resw)given by U(n−w)-derivatives of distributions on Iw, is an isomorphism (ofM∅-representations). Here n−w = w−1nPw ∩ n∅ is the transverse tangentspace of Gw in G≥w.The idea to consider transverse derivatives is not new, and is adopted inmany previous works, e.g. [15], [31]. However the expression of distributionsas transverse derivatives in these reference are on narrow neighbourhoodsand are not explicit enough. In particular, their expressions give no clue onhow to find the distributions D from its restrictions. The expression in theTheorem 6.1 is in a neat algebraic form. More importantly, our expressionsare global on each open G≥w, which will make the globalization of localdistributions easier than the previous works.Torsion Subspaces (Chapter 7)The second crucial idea is to consider the n∅-torsion subspace[(I≥w/I>w)′][n∅•] = [I ′w ⊗ U(n−w)][n∅•]on the dual-quotient (I≥w/I>w)′, where n∅ is the nilpotent radical of thestandard minimal parabolic subalgebra. We will explain the reason to con-sider the torsion subspace at the beginning of Chapter 7. Briefly speaking,in many cases (including the minimal principal series), the image of a mapΦ ∈ HomP∅(V ′, (I≥w/I>w)′) has its image in the n∅-torsion subspace.The space (I≥w/I>w)′ ' Ker(Resw) ' I ′w ⊗ U(n−w) is actually a U(g)-module thus also a U(n∅)-module. The I ′w is also a U(n∅)-module, howeverthe tensor product I ′w ⊗ U(n−w) is not a tensor product of n∅-modules. Thismakes the n∅-torsion subspace on I ′w ⊗ U(n−w) very hard to compute. Sur-prisingly, we have the following main theorem of Chapter 7:Theorem (Theorem 7.1). The n∅-torsion subspace on(I≥w/I>w)′ ' Ker(Resw) ' I ′w ⊗ U(n−w)is given by[I ′w ⊗ U(n−w)][n∅•] = [I ′w][n∅•] ⊗ U(n−w).9This Chapter is the most subtle and technical part of the entire thesis.The above main theorem of Chapter 7 is prove by a combination of geometricand algebraic tricks. It is also one of the innovative part, since no otherreferences have studied the torsion subspaces.Application of Shapiro’s Lemma (Chapter 8)The n∅-torsion subspace[(I≥w/I>w)′][n∅•] ' [I ′w][n∅•] ⊗ U(n−w)is M∅-stable where M∅ is the Levi factor of the minimal parabolic P∅. More-over, the right-hand-side is a tensor product of M∅-representations. To studythe M∅-structure on the torsion subspace [(I≥w/I>w)′][n∅•], we need to studythe M∅-action on the component [I ′w][n∅•], or more precisely on the annihi-lators [I ′w][n∅k] for all k ∈ Z>0. This is the main object in Chapter 8, and itconsists of the following steps:(1) The annihilator-invariant trick: we will show (see (8.1)):[I ′w][n∅k] ' H0(n∅, (Iw ⊗ Fk)′)where Fk is the finite dimensional quotient n∅-module U(n∅)/(n∅k).(2) We will show the Iw = SIndGwP σ is isomorphic to the following Schwartzinduction (Lemma 8.9)Iw ' SIndN∅N∅∩w−1Pwσw,where σw = σ ◦Adw is the twisted representation of N∅∩w−1Pw on V .(3) The tensor product trick: let (ηk, Fk) be the finite dimensional (con-jugation) representation of N∅ on Fk. By combining the above two steps,we will show the following isomorphism (Lemma 8.20):Iw⊗Fk = (SIndN∅N∅∩w−1Pwσw)⊗Fk ' SIndN∅N∅∩w−1Pw(σw⊗ηk|N∅∩w−1Pw).(4) Combining all the above steps, the kth annihilator [I ′w][n∅k] is isomorphictoH0(n∅, [SIndN∅N∅∩w−1Pw(σw ⊗ ηk|N∅∩w−1Pw)]′).We will apply the Shapiro’s Lemma to this n∅-invariant space andshow it is isomorphic toH0(n∅ ∩ w−1pw, V ′ ⊗ F ′k)with the n∅ ∩ w−1pw acts on V ′ ⊗ F ′k by the representation σ̂w ⊗ η̂k.10(5) We will show the M∅ acts on the[I ′w][n∅k] ' H0(n∅ ∩ w−1pw, V ′ ⊗ F ′k)by the representation σ̂w ⊗ η̂k ⊗ γw where γw is a character as in (8.22).In sum, the main result in this chapter is to find the explicit M∅-actionon the annihilators [I ′w][n∅k], namely the Lemma 8.27.Application of the Main Results (Chapter 9)We can combine the main results in Chapter 6, 7, 8, to give a new proofof Theorem 7.2a and Theorem 7.4 in [15] for linear algebraic Lie groups.Let (τ, V ) be a irreducible unitary representation (not necessarily finitedimensional), and σ = τ ⊗ δ1/2P . The I = SIndGPσ (equivalently the normal-ized induction IndGP τ) is irreducible if and only if HomG(I, I) = C.First we can show:Theorem (Theorem 9.8). Let σ = τ⊗δ1/2P be as above. Then the intertwin-ing distributions with supports contained in P are all obtained from scalarintertwining operators by Frobenius reciprocity.From this theorem, we see for unitary (normalized) parabolic inductions,the non-trivial intertwining distributions have their supports containing non-identity double cosets. In particular, for the reducible unitary parabolic in-ductions, the interesting phenomenon occurs on non-identity double cosets.By the above theorem, to show the irreducibility of I, we just need toshow HomP∅(V′, (I≥w/I>w)′) = {0} for all w ∈ [WΘ\W ] and w 6= e.If (τ, V ) is finite dimensional, then the V and V ′ are n∅-torsion. Thenwe haveHomP∅(V′, (I≥w/I>w)′) = HomM∅(V′, [(I≥w/I>w)′][n∅•]).By combining the main theorems in Chapter 6 and 7, we seeHomP∅(V′, (I≥w/I>w)′) = HomM∅(V′, [I ′w][n∅•] ⊗ U(n−w)).Also since V ′ is finite dimensional, there are positive integers k and n (largeenough) such thatHomM∅(V′, [I ′w][n∅•] ⊗ U(n−w)) = HomM∅(V ′, [I ′w][n∅k] ⊗ Un(n−w)).11By the Lemma 8.27, we see the M∅ acts on the [I ′w][n∅k] ⊗ Un(n−w) by therepresentation σ̂w ⊗ η̂k ⊗ γw ⊗ Un(n−w), which is further expressed asτ̂w ⊗ w−1δ−1/2P ⊗ η̂k ⊗ γw ⊗ Un(n−w).Meanwhile, the M∅ acts on V ′ by the representation τ̂⊗δ−1/2P . By comparingthese two M∅-actions, we obtain the following analogue of Theorem 7.4 andTheorem 7.2a in [15]:Theorem (Theorem 9.6). If the representation τ is regular in Bruhat’ssense (see Definition 9.4), then the above two M∅-actions are not equal forall w 6= e, k > 0, n ≥ 0, henceHomP∅(V′, (I≥w/I>w)′) = {0}, ∀w ∈ [WΘ\W ], w 6= e.Therefore the parabolic induction I is irreducible.Theorem (Theorem 9.15). If P = P∅ is the minimal parabolic subgroup,and for all w ∈ W the representation τw is not equivalent to τ , then theabove two M∅-actions are not equal for all w ∈ W,w 6= e, k > 0, n ≥ 0,henceHomP∅(V′, (I≥w/I>w)′) = {0}, ∀w ∈W,w 6= e.Therefore the I is irreducible.The Last Chapter of the ThesisIn the Chapter 10, we will sketch the proof of the general results men-tioned in 1.1.3, i.e. the generalization of the Theorem 6.1, Theorem 7.1and some of the results in Chapter 8. We also discuss some topics we arecurrently working on.1.2 History and MotivationThe parabolic inductions are of great importance in representation the-ory, in the following sense:• It is an important method to construct new (infinite dimensional) rep-resentations from representations of smaller groups.• The subrepresentation theorem, i.e. all irreducible representations oc-curs as a sub or quotient representations in certain parabolic induc-tions. In some sense, they essentially contains all information aboutrepresentations of a group.12• In the character theory, the characters of parabolic inductions form a“dual basis” of the characters of irreducible representations.We summarize some previous study on irreducibility of parabolic induc-tions, which motivate our work in this dissertation.1.2.1 The Work of Bruhat [15]In [15], Bruhat defined the induced representations of Lie groups, andstudied the intertwining forms between two induced representations follow-ing the scheme of Mackey. More precisely, in the above thesis he developedthe theory of quasi-invariant distributions on Lie groups, and applied itto the study of intertwining numbers between two induced representations.Then he applied the results to show certain sufficient conditions of irre-ducibilities for the principal series (non-degenerate and degenerate), inducedfrom parabolic subgroups of connected semisimple Lie groups.In a word, the work of Bruhat gives us a method to estimate the in-tertwining numbers, i.e. it gives upper bound of the intertwining numbersbetween two induced representations ([15] p160 Theorem 6.1 and p171 The-orem 6.3). In particular, it can partially answer the following questions(under strong conditions):• When is a unitary induction irreducible? ([15] p177 Theorem 6.5)• When are two induced representations disjoint?However, it has the following constraint:• The intertwining forms introduced in [15] is different from intertwin-ing operators (see the following subsubsection), and the intertwiningoperators are of more interests to us.• The dual of a unitary induction is isomorphic to the induction of thedual. However, for non-unitary representations of a subgroup, thismay not be true. Hence Bruhat’s method does not apply to the irre-ducibility of non-unitary inductions.• The most important aspect is, the work in [15] only gives estimationof intertwining forms, and it cannot answer what exactly are the in-tertwining forms (operators).These cannot deny the importance of [15], because it is the starting workon induced representations of Lie groups and their irreducibilities. The cru-cial ideas, such as reducing the problem into a distribution analysis, and13the idea to study double cosets separately (also the initial form of “Bruhatdecomposition”), are adopted in most of the subsequent works. In the fol-lowing part of this subsection, we will give a brief review of [15].Remark 1.1. In the entire subsection 1.2.1, we assume all representationspaces are, at least, locally convex Hausdorff topological vector spaces. Andlater in the thesis we will work on the so-called Harish-Chandra representa-tions, and explain why it is sufficient to deal with such representations. Ofcourse for unitary representations there is no ambiguity since representationspaces are all Hilbert spaces.Intertwining Operators and Intertwining Forms (Paragraphe 1-6)The main object studied in [15] is the intertwining forms rather thanintertwining operators:Definition 1.2 ([15] p156). Let G be a Lie group, and for i = 1, 2 let (pii, Ui)be two representations of G.• A continuous linear map T : U1 → U2 is called an intertwiningoperator (from U1 to U2), if pi2(g)◦T = T ◦pi1(g) for all g ∈ G. LetI(U1, U2) = the space of intertwining operators U1 → U2I(U1, U2) = the dimension of I(U1, U2)• A separately continuous bilinear form B : U1 × U2 → C is called anintertwining form (between U1 and U2), if B(pi1(g)u, pi2(g)v) =B(u, v) for all u ∈ U1, v ∈ U2. Leti(U1, U2) = the space of intertwining forms between U1, U2i(U1, U2) = the dimension of i(U1, U2)The i(U1, U2) is called the intertwining number between pi1, pi2.If U1, U2 are unitary representations, then one has ([15] p156)I(U1, U2) = I(U2, U1), and I(U1, U2) = I(U2, U1).However for non-unitary representations, the above equalities are false. (see[15] p156 comment 24.) To overcome this non-symmetric property of inter-twining operators, Bruhat introduce the above notion of intertwining formsand numbers, which is obviously symmetric:i(U1, U2) = i(U2, U1) and i(U1, U2) = i(U2, U1),14for general representations U1, U2 (not only for unitary representations).And Bruhat studied the space of intertwining forms instead of intertwiningoperators in his thesis.Remark 1.3. For unitary representation (pi, U) of G, the pi is irreducible ifand only if I(U,U) = 1. However if (pi, U) is a general representation, thenI(U,U) = 1 only implies pi is indecomposable, but it may not be irreducible.In [15], Bruhat studied the irreducibilities only for unitary inductions. Aswe can see below, for unitary representations, there is no essential differencebetween intertwining operators and forms.Remark 1.4. For i = 1, 2, let (pii, Ui) be two representations of G, andlet ( 2̂, Û2) be the contragredient representations of U2. Then one has theisomorphism between vector spacesi(U1, U2) ' I(U1, Û2)and their dimensions are thus equal:i(U1, U2) = I(U1, Û2).Remark 1.5. Let (σ, V ) be a unitary representation of a subgroup P of G,and let IndGPσ be the normalized induction of σ, and we know the IndGPσ isalso unitary. The IndGPσ is irreducible, if and only if I(IndGPσ, IndGPσ) = 1.By the above discussion, it is irreducible if and only if i(IndGPσ, ÎndGPσ) = 1.As a basic property of unitary inductions, one has ÎndGPσ ' IndGP σ̂, hencethe irreducibility of IndGPσ is equivalent toi(IndGPσ, IndGP σ̂) = 1.Hence the irreducibility problem of unitary parabolic inductions is includedas a special case of the study of intertwining forms between two parabolicinductions. And Bruhat showed the irreducibilities of IndGPσ by showing theupper bound of i(IndGPσ, IndGP σ̂) is 1.Minimal Principal Series (Paragraphe 7, §1− 5)Bruhat proved the following theorem for principal series (induced fromminimal parabolic subgroups):Theorem ([15] p193 Theorem 7.2). Let G be a connected semisimple Liegroup, P∅ be a minimal parabolic subgroup of G with Langlands decomposi-tion P∅ = ◦M∅A∅N∅.15(1) Let (σ, V ) be an irreducible unitary finite dimensional representation ofP∅ (which has to be trivial on N∅ and is extended from a irreducible finitedimensional representation of ◦M∅A∅). If for all non-identity elementw ∈ W of the Weyl group, the representation σ and wσ of ◦M∅A∅are not equivalent, then the (normalized) parabolic induction IndGP∅σ isirreducible.(2) Let (σ1, V1), (σ2, V2) be two irreducible finite dimensional representationsof P∅, then the parabolic inductions IndGP∅σ1 and IndGP∅σ2 are equivalent,if and only if there exists a w ∈W such that the ◦M∅A∅-representationsσ1 and wσ2 are equivalent.We will prove the analogue of part (1) for algebraic Lie groups in Chapter9.Degenerate Principal Series (Paragraphe 7, §6)Let PΘ be a real parabolic subgroup of G corresponding to a subsetΘ of the base of restricted roots, with Langlands decomposition PΘ =◦MΘAΘNΘ. Let (τ, V ) be an irreducible unitary finite dimensional rep-resentation of PΘ, then it is of the formτ = τ◦MΘ ⊗ χΘ ⊗ 1where τ◦MΘ is the restriction of τ to◦MΘ (it is a irreducible unitary repre-sentation of ◦MΘ on V ), χΘ is the restriction of τ to the split componentAΘ (it is a unitary character of AΘ) and 1 means the trivial representationof the unipotent radical NΘ.The restriction of τ to the A∅ is in general not a single character, but afinite set of characters of A∅. LetSpec(A∅, τ) = the A∅-spectrum on (τ, V ).Then this is a finite set of unitary characters of A∅. Let χ ∈ Spec(A∅, τ)be an arbitrary element, then every character in Spec(A∅, τ) are of the formwχ for some w ∈WΘ.Definition 1.6 (Regularity of Spec(A∅, τ)). Consider the following con-dition(Reg-1): wχ 6= χ, ∀w ∈W −WΘ (1.1)If a character χ ∈ Spec(A∅, τ) satisfies the above condition (Reg-1), then wesay χ is regular.It is easy to see if one character in Spec(A∅, τ) is regular, then all char-acters in Spec(A∅, τ) are regular.16Bruhat has shown the regularity is a sufficient condition for irreducibility:Theorem ([15] p203 Theorem 7.4). Let (τ, V ) be an irreducible unitaryfinite dimensional representation of PΘ, and χ ∈ Spec(A∅, τ) be a A∅-character occurring on τ . If χ is regular (in the sense of Definition 1.6),then the (normalized) parabolic induction IndGPΘτ is irreducible.We will reproduce this theorem for algebraic Lie groups in Chapter 9.1.2.2 Some Other Related WorksWe introduce some related works on irreducibilities of unitary parabolicinductions. Most of them are on inductions of finite dimensional unitaryrepresentations (unitary characters or induction from minimal parabolicsubgroups). Our short-term goal is to reproduce these works as much aspossible, then we have the confidence to attack the infinite dimensional(τ, V ).The Work of Wallach on Minimal Principal SeriesWe introduce work of N. Wallach on minimal principal series of splitgroups.The connected complex semisimple Lie groups are all complex algebraicgroups, and by restriction of scalar we can study them as real points groupsof connect semisimple algebraic groups defined over R.Let G be a connected complex semisimple Lie group, with B a Borelsubgroup (minimal parabolic). Assume T is a maximal complex torus ofG, such that B = TN where N is the unipotent radical of B. The Tis isomorphic to a complex torus, hence its irreducible representations arecharacters. Wallach has proved the following strong result:Theorem (Theorem 4.1 on page 112 of [40]). Let G be a connected complexsemisimple Lie group, with B a minimal parabolic subgroup. Let B = TNbe its Levi decomposition, and χ be a unitary character of T . Then thenormalized (Hilbert) parabolic induction IndGBχ is irreducible.Let G = SLn, and G = SL(n,R). Let B be the subgroup of uppertriangular matrices in G, T be the subgroup of diagonal matrices in B, Nbe the unipotent radical of B. The T decomposes intoT = S ×A17whereS = {diag(m1, . . . ,mn) : mi = ±1,m1m2 · · ·mn = 1}A = {diag(a1, . . . , an) : ai > 0, a1a2 · · · an = 1}Let m = (m1, . . . ,mn) be a generic element in S, and let 0 be the trivialcharacter on S: 0(m) = 1,∀m ∈ S. For i = 1, 2, . . . , n − 1, let i be thecharacter i(m) = mi. And let n = 12 · · · n−1. Then every nontrivialunitary characters of S are of the form i1 · · · ir for some 1 ≤ i1 < i2 <. . . < ir ≤ n− 1.Theorem (Theorem 5.1 on page 113 of [40]). Let G = SL(n,R) and let Pbe the minimal parabolic subgroup of upper triangular matrices.• If n is odd, then for any unitary character χ of T , the unitary (Hilbert)principal series IndGBχ is irreducible.• If n is even, and let ξ = i1 · · · ir for some 1 ≤ i1 < . . . < ir ≤ n − 1and r 6= n/2, and let ν be a unitary character of A, then the unitaryprincipal series IndGB(ξ ⊗ ν) is irreducible.• If n is even, and let ξ = i1 · · · ir for some 1 ≤ i1 < . . . < ir ≤ n − 1and r = n/2, and let ν be a unitary character of A, then the unitaryprincipal series IndGB(ξ ⊗ ν) is reducible, and is a direct sum of twoirreducible constituents.Remark 1.7. The results on complex groups are not reachable now. How-ever we are working on the group SL(n,R) and trying to reproduce theabove Theorem on SL(3,C), and find the intertwining operators studied in[37].Degenerate Principal Series for Symplectic GroupsThere are numerous works on the irreducibilities of degenerate principalseries. Some of them use pure algebraic method, by computing the K-typeson the inducted representations. But algebraic methods cannot tell us whatthe intertwining operator is.Among the works on degenerate principal series, the works in [25] and[22], [21] catch our attention.LetG = Sp(2n+ 2,C) = {g ∈ GL(2n+ 2,C) : tgJg = J}18where J is the matrix with all anti-diagonal entries equal to 1, and all otherentries equal to 0. Let M be the subgroupM = {a 0 00 A 00 0 a−1 : a ∈ C×, A ∈ Sp(2n,C)}Then M is the Levi factor of the maximal parabolic subgroup P . Let χ bea unitary character of the torusA = {a 0 00 I 00 0 a−1 : a ∈ C×},and we extend it trivially to the entire M . The unitary induction IndGPχform the unitary degenerate principal series when χ run through all unitarycharacters of A. K. Gross has proved the following resultTheorem (Theorem 7 on page 422 of [25]). If χ is not the trivial characterof A, the IndGPχ is irreducible. If χ is the trivial character, then IndGPχ isreducible with two irreducible constituents.Similar for the complex symplectic groups, T. Farmer has proved ananalogous result. Let G = Sp(2n+2,R) and replace all the above subgroupsP,M,A by subgroups with real matrices, then one hasTheorem (Theorem on page 411 of [22]). If the χ is a non-trivial charac-ter, then IndGPχ is irreducible. If χ is the trivial character, then IndGPχ isreducible with two constituents.Remark 1.8. Actually both Gross and Farmer have computed the commut-ing algebra HomG(IndGPχ, IndGPχ), and shown they are generated by certainMellin transformations on the groups. This motivate us to reproduce theseintertwining operators by using our tools of Schwartz analysis.Parabolic Inductions of Unipotent RepresentationsWe do not have many examples on the irreducibilities of parabolic in-ductions of infinite dimensional representations. Among them, we want tomention the work of Barbasch and Vogan ([4]).Let G be a complex semisimple Lie group, and g be its Lie algebra.Let O be a nilpotent orbit in g, and assume it is special with its dualorbit LO even. Let A(O) be the Lusztig quotient group defined in (4.4c) of19[4]. The irreducible representations of the finite group A(O) parameterize afinite packet of irreducible representations of G, called special unipotentrepresentations (see Theorem III on page 46 of [4]). For each irreduciblerepresentation pi of A(O), let Xpi be the corresponding special unipotentrepresentations.Assume the orbit O is induced from a nilpotent orbit Om of a Levisubalgebra m (thus the O is special), and assume LO is even. Then theLusztig quotient A(O) has a subgroup Am(O). This group is not definedthoroughly in [4], and it is claim to be the same as the group considered inthe book [32] of Lusztig.In the last section of [3], the authors claim the following resultTheorem. Let O, m, Am(O) be as above. Then for each irreducible rep-resentation pi of Am(O), there is an irreducible representation Xmpi of theLevi subgroup M , such that the parabolic induction IndGPXmpi has all its irre-ducible subquotients isomorphic to a representation Xη in the special unipo-tent packet corresponding to O. For each irreducible representation η of theLusztig quotient A(O), the corresponding special unipotent representationXη occurs with multiplicity[η|Am(O) : pi].The representations Xmpi are characterized as follows:(1) they have the infinitesimal characters (wλO, λO) (where λO is definedin (1.15b) in [4]).(2) their left and right annihilators (in U(g)) are maximal among all irre-ducible representations with the same infinitesimal characters.This theorem is prove by matching the characters of the representations,and there is no clue how to find the intertwining operators. It is evenunknown whether the representations Xmpi are unitary.For classical groups, Barbasch has shown in [3] that the index of Am(O)in A(O) is either 1 or 2, hence the induced representation IndGPXmpi has atmost two irreducible factors in the composition series.1.3 Reading Guide and Key Points of theChapter 2, 3 and 4The entire thesis is divided into two parts: the foundation part consistsof Chapter 2, 3 and 4. The main part consists of Chapter 5, 6, 7, 8, 9. The20main part is already summarized at the beginning of the thesis. Here wewrite a short reading guide for the foundation part.We recall necessary notions and results, and omit all details by givingprecise references. The Chapter 2, 3 recall some basic knowledge whichmay be well-known to the reader, while the stuff in Chapter 4 are some notwell-known works from [1], [2] and [20], and our own works.1.3.1 Chapter 2The Chapter 2 is short review on three topics: algebraic Lie groups,topological vector spaces and representations of Lie groups. The section2.1 and 2.3 are well-known and we keep them in the shortest form. Thenotions of NF/DNF-spaces are not so well-known and the reader need topay attention to the section 2.2.• The Lie groups studied in the thesis are real point groups of con-nected reductive linear algebraic groups defined over R. Thesection 2.1 is a short summary of the structure theory on algebraicgroups and their real point groups. We mainly follow the book [10] ofBorel.• We will mainly work on a particular class of topological vector spaces:nuclear TVS or more precisely NF/DNF-spaces. All terms and resultscould be found in Treves’ [36], and for the properties of NF/DNF-spaces, we follow the appendix in [18]. The NF/DNF-spaces are verysimilar to finite dimensional vector spaces, they behave well under al-gebraic construction (e.g. strong dual and tensor products are exact),and these good algebraic properties in crucial to build the distributiontheory in Chapter 4.• The representations studied in the thesis are Harish-Chandra rep-resentations. The section 2.3 is a short review of the notion ofHarish-Chandra representations introduced in [17]. We also give aquick review of the notions of smooth inductions, Hilbert inductionsand Frobenius reciprocity.1.3.2 Chapter 3The Chapter 3 consists of three parts:• In 3.1 and 3.2, we recall basic notions in real algebraic geometry. Wefollow the [6] in section 3.1, and the [1], [2] in section 3.2.21The section 3.1 is a quick review of affine real algebraic varieties. Typ-ical examples of affine real algebraic varieties are G(R) for algebraicgroup G and double cosets on G(R) of algebraic subgroups.In 3.2, we recall the notions of (affine) Nash manifolds, on which thetheory of Schwartz functions is built in [1]. We basically follow thedefinitions from [1]. The nonsingular affine real algebraic varieties isthe most important class of affine Nash manifolds, and we will onlyconsider affine real algebraic varieties in the thesis.• In 3.3, we study the double coset decomposition on the G(R). Theimportant concepts are anti-actions, closure order, and the notationsG≥i, G>i.• In 3.4, we study the torsion subspace of tensor product module overa Lie algebra. This is a pure algebraic section, and it could be readindependently. The reader can skip the last four subsections (from3.4.3 to 3.4.7), and they will not affect the reading.Let h be a complex Lie algebra, and M1,M2 be two (left) h-modulesandM1⊗M2 be their tensor product modules, we prove the followingresults:– M[h•]1 ⊗M[h•]2 ⊂ (M1 ⊗M2)[h•].– If one of them, say M2 is finite dimensional and h-torsion, i.e.M[h•]2 =M2, then the above inclusion is equality:M[h•]1 ⊗M[h•]2 = (M1 ⊗M2)[h•].These results sounds elementary but still not found in reference. Weinclude them for the reader to see how much harder the Chapter 7 is.1.3.3 Chapter 4Some parts of this chapter is also innovative, and not in any reference,but it is not hard to derive all results in this chapter from the knowledge in[1], [2] and [20].The Chapter 4 has two parts:• In section 4.1, 4.2, 4.3 and 4.4, we recall the work of Aizenbud-Gourevitch in [1] and [2]. In these sections, the geometric objectsare affine Nash manifolds.22– In 4.2, we recall the notion of Schwartz R-valued functions onaffine Nash manifolds, and crucial properties on the spaces ofSchwartz functions (e.g. NF-space, cosheaf structure etc). Thenotions and results are generalized to C-valued Schwartz func-tions without difficulty. We will only consider complex valuedSchwartz functions (or even vector valued Schwartz functions incomplex TVS), and the analogous properties are summarized inLemma 4.21.– In 4.3, we generalize the definitions of scalar valued Schwartzfunctions to E-valued Schwartz functions where E is a NF-space.The E-valued Schwartz function spaces have similar properties asscalar valued case, and we summarize the analogous propertiesas Proposition 4.30.– In 4.4, we study the strong dual spaces of Schwartz functionspaces and call elements in them Schwartz distributions. Thecrucial properties of Schwartz distribution spaces are summarizedin Lemma 4.32, Lemma 4.35, Lemma 4.36. And the sheaf prop-erty of Schwartz distributions lead to the following crucial result:the space of Schwartz distributions supported in a closed Nashsubmanifold is independent of the neighbourhood (Lemma 4.38).This result plays an important role in Chapter 6.• In section 4.5 and 4.6, we will work on nonsingular affine real alge-braic varieties. Such varieties is a special class of affine Nash mani-folds that we will only study in the thesis. On nonsingular affine realalgebraic varieties, the Schwartz functions defined in [1] coincide withthe Schwartz functions defined in [20]. Currently B. Elazar and A.Shaviv have generalized the work of [1] to non-affine/non-smooth realalgebraic varieties.– In 4.5, we first restrict the cosheaf of Schwartz functions andsheaf of Schwartz distributions to the Zariski topology on non-singular affine real algebraic varieties, and call them pseudo-cosheaf/pseudo-sheaf on the Zariski topology. We show that aSchwartz distribution has a well-defined support under the Zariskitopology. When the spaces of Schwartz functions are built on al-gebraic groups, these spaces have the right regular actions onright stable subvarieties.– In 4.6, we combine the notion of (local) Schwartz inductions in-troduced in [20], with the properties of Schwartz functions devel-23oped in [1]. More precisely, the (local) Schwartz induction spacesare defined to be the images of Schwartz function spaces undercertain mean value integration map. In particular, the Schwartzinduction spaces are NF-spaces and behave well under the ZariskiP -topology. Therefore we can develop a set of crucial propertiessimilar to the Proposition 4.30 on Schwartz functions, and thedistribution analysis on Schwartz inductions is very much similarto the analysis on Schwartz functions. Note that the Schwartz in-duction from parabolic subgroups are exactly smooth inductionssince the base quotient manifold is compact.1.4 Convention on NotationWe give a short index of notations frequently used in the thesis.Algebraic Groups and Lie GroupsWe will fix the notations of algebraic groups and Lie groups in 2.1. Inthe entire thesis, we follow the “notation-choosing rules” in Remark 2.1.From Chapter 6, we use the following notations of subgroupsNw = w˙−1NP w˙N+w = w˙−1NP w˙ ∩N∅= Nw ∩N∅N−w = w˙−1NP w˙ ∩N∅= Nw ∩N∅Topological Vector SpacesLet V, V1, V2 be topological vector spaces (TVS for short)V ′ = the strong topological dual of a TVS VV ∗ = the algebraic dual of VL(V1, V2) = the space of continuous linear maps from V1 to V2V1 ⊗̂ V2 = the completed tensor product of V1 and V2Representation TheoryLet (σ, V ) be a continuous representation of a Lie group G, then wedenote by (σ̂, V ′) the contragredient (dual) representation.24For two representations (σ1, V1), (σ2, V2) of a Lie group, we denote byHomG(V1, V2) = HomG(σ1, σ2)the space of intertwining operators from (σ1, V1) to (σ2, V2).Let P be a real parabolic subgroup of G (see 2.1), and (τ, V ) be a Hilbertrepresentation of P , thenC∞IndGP τ = the smooth parabolic induction (Definition 2.30)IndGP τ = the normalized Hilbert parabolic induction (Definition 2.33)SIndGP τ = the Schwartz induction (Definition 4.57)Lie Algebras and Modules (Section 3.4 and Chapter 7)All real Lie algebras are denoted by fraktur letters with a subscript0, e.g.g0, and its complexification is denoted by the same letter with thesubscript removed, e.g. g.For a complex Lie algebra h, letU(h) = the universal enveloping algebra(hk) = the two-sided ideal generated by k-products of element in hUn(h) = the finite dimensional subspace of U(h)spanned by i-products of elements in h, i ≤ nLet M be a left h-module (left U(h)-module), we letM[hk] = the annihilator of the ideal (hk),∀k ≥ 0M[h•] =⋃k≥0M[hk]= the (left) h-torsion submodule of MSimilarly for M be a right h-module (right U(h)-module), we letM[hk] = the annihilator of the ideal (hk), ∀k ≥ 0M[h•] = the (right) h-torsion submodule of MLie Algebra Elements as Vector Fields (Chapter 7)Let G be a Lie group, and g0 be its abstract (real) Lie algebra. Anelement X ∈ g0 could be regarded as a left invariant vector field on G,25denoted by XL, and as a right invariant vector field on G, denoted by XR.For a point g ∈ G, let XLg (resp. XRg ) be the tangent vector of XL (resp.XR) in the tangent space TgG at g. We only need these notations in Chapter7.Double Cosets on GLet G be a connected reductive linear algebraic groups defined over R,let PΘ,PΩ be two standard parabolic R-subgroups corresponding to sub-sets Θ,Ω of the base. Let WΘ,WΩ be the corresponding parabolic Weylsubgroups. We use the following notations (from [16]):[W/WΩ] = {w ∈W |wΘ ⊂ Σ+}[WΘ\W ] = {w ∈W |w−1Θ ⊂ Σ+}[WΘ\W/WΩ] = [W/WΩ] ∩ [WΘ\W ]The [WΘ\W/WΩ] is the set of minimal representatives (see 3.3), and it is inone-to-one correspondence with the (PΘ, PΩ)-double cosets in G.By abuse of notation, we use w to denote both the element in W and afixed representative of it in G. For every w ∈ [WΘ\W/WΩ], we denote byGΩw the double coset PΘwPΩ. Under the closure order defined in 3.3, we letGΩ≥w =∐PΘwPΩ⊂PΘxPΩPΘxPΩGΩ>w = GΩ≥w −GΩwThe GΩ≥w and GΩ>w are Zariski open in G, while the GΩw is closed in GΩ≥w.In particular, when Ω = ∅ (empty set), we simply drop all superscript.Schwartz Function Spaces and Schwartz InductionsLet X be a nonsingular real affine algebraic variety (see 3.1), or moregenerally an affine Nash manifold (see 3.2), let E be a nuclear Fre´chet space.Then we denote byS(X,R) = the space of R-valued Schwartz functions on XS(X,C) = the space of C-valued Schwartz functions on XS(X,E) = the space of E-valued Schwartz functions on X26Let P be a real parabolic subgroup of G, σ be a nuclear Fre´chet rep-resentation of P , which is of moderated growth, let Y be a left P -stablesubvariety of G, then we letSIndYP σ = the local Schwartz induction space (Definition 4.64).In particular, for P = PΘ, and Ω ⊂ Θ, we use the following simplifiednotations for local Schwartz inductions:IΩw = SIndGΩwP σIΩ≥w = SIndGΩ≥wP σIΩ>w = SIndGΩ>wP σAs before, when Ω = ∅ (empty set), we omit the superscript Ω, and for eachw ∈ [WΘ\W ], we letIw = I∅wI≥w = I∅≥wI>w = I∅>w27Chapter 2Basic SettingSummary of This ChapterIn this chapter, we unify the terminology and notations, and give precisereference for the key results without giving proof.• In 2.1, we recall the basic notions and results on linear algebraicgroups. The Lie groups studied in this thesis are real points of con-nected reductive linear algebraic groups defined over R.• In 2.2, we recall the basic notions of nuclear Fre´chet (NF)-spaces andtheir strong dual (DNF-spaces). These topological vector spaces forma good category similar to the category of finite dimensional vectorspaces, and they behave well under algebraic constructions, e.g. thestrong dual and tensor product functors are exact, and tensor productscommute with strict direct limit. These good properties are crucial inthe study of Schwartz functions/inductions and distributions (whichwill be introduced in Chapter 4).• In 2.3, we recall the definition of Harish-Chandra representations in-troduced in [17], and the main object of the thesis—smooth parabolicinductions.2.1 Groups Studied in the ThesisThe groups we will study in this thesis are groups of real points of con-nected reductive linear algebraic groups defined over R.For algebraic groups, we follow the terms and notations from the follow-ing book and papers of Borel: [7], [8], [11], [10]. In particular, all algebraicgroups are treated in the classical sense, and regarded as complex algebraicvarieties. In terms of group schemes, the algebraic groups studied in this the-sis are R-group schemes, and we identify them with their C-rational points,and study the groups of R-rational points of them.28• The 2.1.1 is on the relative theory of reductive linear algebraic groupsdefined over R. We recall the relative root datum, the structure ofparabolic R-subgroups and relative Bruhat decompositions on suchgroups.• The 2.1.2 is on Lie groups. The real points groups of connected reduc-tive linear algebraic groups are of Harish-Chandra class. The algebraicstructure theory coincide with the Lie group structure theory on suchgroups.2.1.1 Algebraic Groups in this ThesisLet G be a connected reductive linear algebraic group defined over R.Let G(R) be the group of R-rational points of G, then G(R) has a smoothLie group structure.We denote the Lie group G(R) by G, and letg0 = LieGbe its (real) Lie algebra. Let gC be the algebraic Lie algebra of the G, theng0 is a real form of gC.Remark 2.1. Throughout the entire thesis, we stick to the following rulesof notations: for an algebraic group (defined over R) denoted by a boldfaceletter, (e.g. H),• its (complex) algebraic Lie algebra is denoted by fraktur letter withsuperscript C (e.g. hC);• its Lie group of real points is denoted by the same uppercase letter(e.g. H);• the Lie algebra of the real point Lie group is denoted by fraktur letterwith a subscript 0 (e.g. h0);• the complexification of h0 is denoted by the same fraktur letter withoutsubscript (e.g. h).29Relative Roots, Relative Weyl GroupLetS = a maximal R-split torus of GΣ = the set of S-roots(relative roots)NG(S) = the normalizer of SZG(S) = the centralizer of SW = NG(S)/ZG(S)= the relative Weyl groupFor each α ∈ Σ, let gCα be the α-root subspace on gC. In general (unlike theabsolute theory), the gCα is neither one dimensional, nor a Lie subalgebra.If 2α /∈ Φ, then gCα is a Lie subalgebra of gC. If 2α ∈ Φ, then gCα + gC2α is asubalgebra of gC. LetΦred = {α ∈ Φ : 2α /∈ Φ}be the subset of reduced roots. For each α ∈ Φred, let (α) = {kα ∈ Φ|k ∈Z+} andgC(α) ={gCα if α ∈ ΦredgCα + gC2α if α /∈ Φred(2.1)Then one has the Lie algebra decomposition ([10] p231 §21.7):gC = LieZG(S)⊕⊕α∈ΦgCα = LieZG(S)⊕⊕α∈ΦredgC(α). (2.2)For each α ∈ Φ, there exists a unique closed connected unipotent R-subgroup U(α) normalized by ZG(S) with Lie algebra gC(α) ([10] p232 Propo-sition 21.9). It is the unipotent subgroup directly spanned by absolute rootsubgroups Uβ such that β ∈ ΦC restricted to S equals to α.Standard Parabolic R-subgroupsWe fix a minimal parabolic R-subgroup P∅ containing S, it determinesthe following datum on the relative root system:Σ+ = the relative roots occurring in the Lie algebra p∅C∆ = the base of Σ determined by Σ+Σ− = −Σ+S = {sα : α ∈ ∆}30The Σ+ is a positive system of Σ which determines the base ∆, and the Sis a generator set of W .The parabolic R-subgroups containing P∅ are called standard parabolicR-subgroups, and they are in one-to-one correspondence with subsets of ∆.For each subset Θ ⊂ ∆, letSΘ = (⋂α∈ΘKerα)◦MΘ = ZG(SΘ)NΘ = 〈U(α) : α ∈ Σ+ − 〈Θ〉〉WΘ = 〈sα : α ∈ Θ〉Then the PΘ = P∅ ·WΘ ·P∅ is a standard parabolic R-subgroup with LeviR-factor MΘ, unipotent radical NΘ, and Levi decomposition MΘ n NΘ.The WΘ is the relative Weyl group W (MΘ,S) of MΘ.In particular when Θ = ∅, the parabolic R-subgroup of G correspondingto ∅ ⊂ ∆ is exactly the P∅ we fixed at the beginning, and this is why weuse the subscript ∅. And the S∅ = S, W∅ = {e} (identity).The Relative Bruhat decompositionLet G(R),P∅(R),NG(S)(R) be the groups of R-rational points on G,P∅and NG(S), and S = {sα : α ∈ ∆} be the set of reflections of R-simple roots.Then the quadruple(G(R),P∅(R),NG(S)(R), S)is a Tits system ([10] p236 Theorem 21.15).Let Θ,Ω be two subsets of ∆, and PΘ,PΩ the corresponding standardparabolic R-subgroups, and WΘ,WΩ be the corresponding (parabolic) sub-groups of W . By the same remark on p22 of [13], one has the double cosetsdecomposition of G(R):G(R) =∐w∈[WΘ\W/WΩ]PΘ(R)wPΩ(R). (2.3)Here we still adopt the notation of the representative sets on page 7 of[16]:[W/WΩ] = {w ∈W |wΩ ⊂ Σ+}[WΘ\W ] = {w ∈W |w−1Θ ⊂ Σ+}[WΘ\W/WΩ] = [W/WΩ] ∩ [WΘ\W ]312.1.2 G as a Lie GroupLet G be a connected reductive linear algebraic group defined over R,and G = G(R) be the Lie group corresponding to the group of real rationalpoints. We assume G has nonzero semisimple R-rank, i.e. the R-rank ofDG is nonzero, to make sure G has proper parabolic R-subgroups. For Gand its R-subgroups, we follow the notation-choosing rules in Remark 2.1.The G has the following properties:1. G has finite number of connected components (under the Euclideantopology).2. Its Lie algebra g0 = LieG is reductive.3. Let G◦ be its identity component, then the derived subgroup D(G◦)has finite center and is closed in G◦.4. The G is of inner type, i.e. AdG ⊂ Intg where g ' LieG.Hence the G is a Lie group of Harish-Chandra class in the sense of [28]p105-106. Also, the G is a real reductive Lie group in the sense of [12].Cartan Decompositions and Restricted RootsSince G has finitely many components, every compact subgroup is con-tained in a maximal one. Every maximal compact subgroup meets all con-nected components, and they are conjugated under G◦. There exists aglobal Cartan decomposition on G: G is diffeomorphic to a productK × V where K is an arbitrary maximal compact subgroup and V is a vec-tor subgroup. The corresponding involution Θ : k · v 7→ k · v−1 is called aglobal Cartan involution of G ([33] section 3).The differential θ = dΘ of the global Cartan involution is called the localCartan involution on the Lie algebra g0, which gives the local Cartandecompositiong0 = g0(θ, 1)⊕ g0(θ,−1)where g0(θ,±1) are the ±1-eigenspaces of the involution θ. The eigenspaceg0(θ,+1) is exactly the Lie algebra k0 of K.For the −1-eigenspace, a subspace of g0(θ,−1) is a subalgebra of g0 ifand only if it is abelian. All maximal abelian subalgebra of g0(θ,−1) areconjugate under K◦.Let a0 be a maximal abelian subalgebra of g0(θ,−1). The adjoint actionof a0 on g0 gives weight space decomposition. For λ ∈ a∗0 (real dual), one32has the weight space g0λ = {X ∈ g0|[Z,X] = λ(Z)X,∀Z ∈ a0}. The λ thatg0λ 6= {0} are called a0-roots or restricted roots on g0. LetΣ(g0, a0) = the set of a0-roots on g0W (g0, a0) = NK◦(a0)/ZK◦(a0)= the restricted Weyl groupThen the Σ(g0, a0) form a root system in the vector space it spans withW (g0, a0) as its Weyl group.Restricted Root System vs Relative Root SystemAs in 2.1.1, by fixing a maximal R-split torus S of G, one has the rela-tive root system Σ = Σ(G,S) and the relative Weyl group W = W (G,S)determined by S.As above, by fixing a Cartan involution Θ on G (θ on g0), and a max-imal abelian subalgebra a0 in g0(θ,−1), one has the restricted root systemΣ(g0, a0) and the restricted Weyl group W (g0, a0) determined by a0.We can choose the S, Θ and a0 to be compatible. More precisely, let Sbe the Lie group of the S(R). The Lie algebra s0 of S (which is also theLie algebra of A) is conjugate to the a0 we fixed above. In particular, wecan choose S and a0 such that the s0 = a0. In this case, the A∅ = exp a0 isexactly the identity component S◦.The restriction of the differential α 7→ dα 7→ dα|a0 thus gives a bijectionbetween the relative root system and restricted root system:Σ(G,S)→ Σ(g0, a0) (2.4)α 7→ dα|a0By abuse of notation, we use the same α to express both the algebraic rootin Σ(G,S) and the corresponding restricted root in Σ(g0, a0). The abovebijection also induces an isomorphism between the relative and restrictedWeyl groups: W (G,S) 'W (g0, a0).Real Parabolic Subgroups.A subgroup of G is called a (real) parabolic subgroup of G if it is thereal points of a parabolic R-subgroup of G. In particular, we call P = P(R)a minimal (real) parabolic subgroup of G, if the P is minimal parabolicR-subgroup of G.33As in 2.1.1 we have fixed a minimal parabolic R-subgroup of G contain-ing S, denoted by P∅, then the P∅ = P∅(R) is a minimal parabolic subgroupof G. Similar to the algebraic parabolic subgroups, we call a (real) parabolicsubgroup P of G standard, if the corresponding P is standard, i.e. con-taining the P∅. Thus by fixing the minimal parabolic R-subgroup P∅, thestandard parabolic R-subgroups of G are in one-to-one correspondence withthe standard (real) parabolic subgroups of G, and the correspondence isP 7→ P(R). And the standard (real) parabolic subgroups of G are alsoparameterized by subsets of ∆.For a standard parabolic R-subgroup P = PΘ, its unipotent radicalNP = NΘ and a Levi R-factor MP = MΘ are given in 2.1.1. The two factorsare defined over R, and the semi-direct product induces the semi-directproduct decomposition on P(R): PΘ(R) = MΘ(R)nNΘ(R). Following thenotation-choosing rules in Remark 2.1, we have the Lie group semi-directproduct:PΘ = MΘ nNΘ.And we call this decomposition the Levi decomposition of PΘ, and callthe MΘ a Levi factor of PΘ and NΘ the unipotent radical of PΘ.The Langlands Decomposition of PΘThe R-torus SΘ = (⋂α∈Θ Kerα)◦ is a subtorus of S. The Lie groupidentity component AΘ = SΘ(R)0 is a subgroup of A∅ = S(R)0. (Notethat S = S∅). It is connected, simply connected and isomorphic to directproducts of R>0. It is abelian and the its exponential map is a smoothdiffeomorphism.The AΘ has a unique complement inside the group MΘ, namely the◦MΘ :=⋂χ∈X(MΘ)Ker|χ|,here X(MΘ) = Homcont(MΘ,R×) is the group of continuous homomor-phisms from MΘ to R×. The ◦MΘ is in general not connected, but is areal reductive Lie group or a group of Harish-Chandra class. If Θ is theempty set, the ◦MΘ is a compact group.And the MΘ is a direct product of the two subgroups:MΘ = AΘ × 0MΘ.The decompositionPΘ =0MΘ ×AΘ nNΘ34is called the Langlands decomposition of the real parabolic subgroup PΘ.The AΘ and◦MΘ are not real point groups of algebraic groups. But westill follow the notation-choosing rules in Remark 2.1, and let aΘ0 = LieAΘ(real Lie algebra) and aΘ be its complexification.2.2 NF-Spaces and DNF-SpacesWe recall some crucial properties of nuclear Fre´chet spaces and theirdual. For basic terms and results, we follow the book [36]. Most of thematerials about NF-spaces and DNF-spaces are from the short survey inthe Appendix A of [18].2.2.1 Basic Notions on Topological Vector SpacesIn this thesis, the term topological vector space(s) is abbreviated asTVS. If not otherwise stated, all TVS are assumed to be locally convex andHausdorff.• A TVS is Fre´chet ([36] p85), if it is locally convex, metrizable andcomplete.• A locally convex Hausdorff TVS E is nuclear, if for every continuousseminorm p on E, there is a continuous seminorm q such that p ≤ qand the canonical map Eˆq → Eˆp is nuclear ([36] p479 Definition 47.3).Definition 2.2. A nuclear Fre´chet TVS is called an NF-space for short,and the strong dual of an NF-space (with the strong topology) is called aDNF-space.The strong dual of a NF-space is also nuclear ([36] p523 Proposition50.6), but almost never Fre´chet.Homomorphisms and Topological Exact Sequences of TVSLet E,F be two TVS, and φ : E → F be a continuous linear map. Asin linear algebra, it factor through the linear map between vector spaces:φ0 : E/Ker(φ)→ F.which is called the map associated with φ. With the quotient topology onE/Ker(φ), the φ0 is continuous. It has a naive linear inverse φ−10 : φ(E)→E/Ker(φ) since φ0 is a linear isomorphism onto the φ(E), but this inverse35φ−10 need not to be continuous. The φ−10 is continuous if and only if φ0 isan open map, or equivalently the quotient topology on φ(E) from φ0 is thesame as the induced topology from F .Definition 2.3 ([36] p35 Definition 4.1). A continuous linear map φ : E →F is called a homomorphism of TVS, (or strict morphism) if its asso-ciated map φ0 is open.Lemma 2.4 ([36] p170 Theorem 17.1). Let E,F be two metrizable andcomplete TVS. Then every surjective continuous linear map φ : E → F is ahomomorphism.Lemma 2.5 ([34] p77 Corollary 1). Let E,F be two Fre´chet spaces and φ :E → F be an injective continuous linear map. Then φ is a homomorphismif and only if its image φ(E) is closed in F .Definition 2.6. Let E1, E2, E3 be three TVS, then the sequence0→ E1 → E2 → E3 → 0is called an (short) exact sequence of TVS if all maps are continuouslinear map and it is an exact sequence in the category of vector spaces.Furthermore, it is called a (short) topological exact sequence ofTVS, if all maps are homomorphisms of TVS. Equivalently this meansE1 → E2 and E2 → E3 are homomorphisms of TVS, since the other two areautomatically homomorphisms.Lemma 2.7. Any short exact sequence of Fre´chet spaces (in the categoryof vector spaces) is automatically topological exact, i.e. all maps are homo-morphisms.Proof. Let E1, E2, E3 be three Fre´chet spaces, and let0→ E1 → E2 → E3 → 0be an exact sequence in the category of vector spaces. Then by Lemma 2.4,the E2 → E3 is a homomorphism. Also since Im{E1 → E2} = Ker{E2 →E3} which is a closed subspace of E2, by Lemma 2.5, the E1 → E2 is also ahomomorphism.2.2.2 Properties of NF-Spaces and DNF-SpacesThe category of NF-spaces (or DNF-spaces) with homomorphisms be-tween them form a good category, which is similar to the category of finitedimensional vector spaces. We are especially interested in the dual exactsequences, and topological tensor products.36Strong Dual and Transpose MapsLemma 2.8 ([18] p186 Lemma A.2). Let (C•, d•) be a complex of NF-spaces(resp. DNF-spaces), i.e. one has a complex. . .dn−1−−−→ Cn dn−→ Cn+1 dn+1−−−→ . . .each space Cn is a NF-space (resp. DNF-space), and each differential dn isa homomorphism of TVS. Then the strong dual complex. . .dtn+1−−−→ (Cn+1)′ dtn−→ (Cn)′ d′n−1−−−→ (Cn−1)′ → . . .is a complex of DNF-spaces, i.e. all strong dual spaces (Cn)′ are DNF-spaces, and all transpose maps dtn are homomorphisms of TVS. Moreoverwe have the isomorphismHp(C ′•) = H−p(C•)′.In partiular, we haveLemma 2.9. Let0→ E1 φ−→ E2 ψ−→ E3 → 0be a topological exact sequence of NF-spaces (resp. DNF-spaces), then itsstrong dual0→ E′3 ψt−→ E′2 φt−→ E′1 → 0is a topological exact sequence of DNF-spaces (resp. NF-spaces).Topological Tensor ProductsLet E be a nuclear TVS, and F be an arbitrary locally convex HausdorffTVS. Then the inductive topology and the projective topology on E ⊗ Fare the same:E ⊗ F = E ⊗pi Fhence they have the same completions:E ⊗̂ F = E ⊗̂pi F.When one of the TVS in the tensor product is nuclear, we will notdistinguish the inductive and projective tensor products, and simply denotethe tensor product by E ⊗ F , and its completion by E ⊗̂ F .37Lemma 2.10. Let E,F be two locally convex Hausdorff TVS.• If E,F are both nuclear, then so is E ⊗̂ F . ([36] p514 Proposition50.1)• If E,F are both NF-spaces, then so is E ⊗̂ F . ([26] Ch I, §1, no.3Prop 5)• If E,F are both DNF-spaces, then so is E ⊗̂ F . ([26] Ch I, §1, no.3Prop 5)([26] Ch I, §1, no.3 Prop 5)Lemma 2.11. Let E,F be two Fre´chet spaces. If E is nuclear, then• the canonical map E′⊗F ′ → (E ⊗̂F )′ extends to an isomorphism ([36]p524 Proposition 50.7)E′ ⊗̂ F ′ ∼−→ (E ⊗̂ F )′ (2.5)• the canonical map E′⊗F → L(E,F ) extends to an isomorphism ([36]p525 (50.17))E′ ⊗̂ F ∼−→ L(E,F ) (2.6)Lemma 2.12 ([18] p187 Lemma A.3). Let F be an NF-space (resp. DNF-space), let0→ E1 → E2 → E3 → 0be a topological exact sequence of NF-spaces (resp. DNF-spaces). Then0→ E1 ⊗̂ F → E2 ⊗̂ F → E3 ⊗̂ F → 0is a topological exact sequence.Direct Limits and Tensor ProductsLet {Fi : i ∈ I} be an inductive system of nuclear spaces, and let F =lim−→i∈I Fi be its inductive limit in the category of locally convex spaces. IfI is countable and F is Hausdorff, then F is nuclear by [26] Ch II §2 no.2Corollary 1. Moreover, if Fi are DNF-spaces, then so is F = lim−→i∈I Fi.Lemma 2.13 ([18] p187 Lemma A.4). Let E be a DNF-space, and {Fi :i ∈ I} be a inductive system of DNF-spaces. Assume I is countable andF = lim−→i∈I Fi is Hausdorff. ThenE ⊗̂ F = lim−→i∈I(E ⊗̂ Fi). (2.7)382.3 Representations Studied in the ThesisWe introduce the category of representations studied in the thesis, i.e.the Harish-Chandra representations.Throughout this section, let G = G(R) be the Lie group of real rationalpoints of a linear algebraic group G defined over R.2.3.1 Basic Notions on RepresentationsIn literatures, people add various of restrictions on the representationspaces. For example, unitary representations are on Hilbert spaces. We willstudy the intertwining operators and irreducibility, which is a very generalproblem. Thus it is natural to start from spaces with minimal confinement.Continuous RepresentationsA TVS is• locally convex ([36] p58 Definition 7.2), if its topology is defined bya family of seminorms.• Hausdorff, if it is separated (T2) in the ordinary sense , i.e. everytwo points are separated by two disjoint open subsets.• quasi-complete ([14] III.6 Definition 6), if it is locally convex andevery closed and bounded subset is complete.This is the basic assumptions on TVS. In this thesis, if not otherwise stated,all TVS are assumed to be locally convex Hausdorff and quasi-complete.Definition 2.14. A continuous representation (pi, V ) of G (V satisfyingthe above conditions), is a group representation on the TVS V such thatthe mapG× V → V, (g, v) 7→ pi(g)vis continuous (under the product topology).Example 2.15. The left and right regular representation of G on the spaceC∞(G,C) or C∞c (G,C).Remark 2.16. The early works on Lie group representations usually havephysics backgrounds, and the representations under concern are mostly uni-tary (Hilbert) representations, e.g. the work [5] of Bargmann. A lot ofsubsequent references define general representations on Hilbert spaces, for39example the extensive works by D. Vogan including the standard reference[38], and “textbooks” like [41] and [30].Harish-Chandra is one of the founders of algebraic study of Lie group rep-resentations. His early works in 1950’s are on Banach representations, e.g.the “Representations of semisimple Lie groups I-VI”. The groups in his earlyworks are connected semisimple Lie groups. The work of representations onnormed spaces traces back to [23] and [24]. In [27], Harish-Chandra startedto work on representations on locally convex TVS (p5 §2) as in [15], and in[28], he generalized the groups under study to “groups of Harish-Chandraclass”.By tracing back the reference chain, the French school seems to be theorigin of studying representation theory on general TVS, by regarding Liegroups as locally compact topological groups (e.g.[15], [9] and [12]. Theconditions “locally convex, Hausdorff and quasi-complete” on TVS originatefrom these references. The monograph [42] also adopt the general definitionof representations on TVS, but in addition he assumed the representationspace to be complete.Smooth Vectors and K-Finite VectorsDefinition 2.17. Let (pi, V ) be a continuous representation of G. A vectorv ∈ V is called a smooth vector, if the mapG→ V, g 7→ pi(g)vis a smooth V -valued function on G. We denote the subspace of smoothvectors on V byV sm.A continuous representation (pi, V ) is called a smooth representation ifV = V sm.Let K ⊂ G be a maximal compact subgroup, and (pi, V ) be a continuousrepresentation of G.Definition 2.18. A vector v ∈ V is called a K-finite vector, if it iscontained in a finite dimensional K-stable subspace of V . We denote byV Kthe subspace of K-finite vectors on V .All maximal compact subgroups of G are conjugate, hence the abovenotion of “K-finiteness” is independent of the choice of K.40Representations of Moderate GrowthWe fix an embedding G ⊂ GLn defined over R. Therefore the real pointgroup G = G(R) ⊂ GL(n,R) is a group of real matrices, with the action onthe vector space Rn.Definition 2.19 ([39] 1.6). Let the matrix group GL(n,R) act on Rn⊕Rnbyg · (x1, x2) := (g · x1, tg−1 · x2),and let |g| be the operator norm of the above g as an operator on Rn ⊕Rn.We define the norm || · || on G = G(R) ⊂ GL(n,R) by||g|| := |g|.Definition 2.20 ([17] p391). A continuous representation (pi, V ) of G is ofmoderate growth, if(T1) the V is a Fre´chet space, and assume its topology is defined by thefamily P of seminorms;(T2) for each seminorm ρ ∈ P, there is another seminorm ρ′ ∈ P and apositive integer N such thatρ(pi(g)v) ≤ ||g||Nρ′(v)for all g ∈ G, v ∈ V (the N doesn’t depend on g and v).Remark 2.21. If we choose a different embedding G ⊂ GLm, we obtainanother norm on G. The new norm is “equivalent” to the original one, inthe sense that they are bounded by powers of each other. Hence the abovenotion “moderate growth” is independent of the choice of the embeddingG ⊂ GLn.Remark 2.22. The above norm ||·|| : G(R)→ R is bounded by an algebraicfunctions. Hence the above notion of “moderate growth” is equivalent tothe notion “croissance mode´re´e” in Definition 1.4.1 on page 272 of [20].Actually a wide class of representations (including all finite dimensionalcontinuous representations) are of “moderate growth”:Lemma 2.23 ([39] p293 §2.2 Lemma). Every Banach representation is ofmoderate growth.412.3.2 Harish-Chandra ModulesThe Notion of (g, H)-ModuleLet g be the complexified Lie algebra of G, and H ⊂ G be a closedsubgroup of G.Definition 2.24 ([17] p393). A vector space V is called a (g, H)-module,if one has an action of H on V and an g on V , satisfying the followingconditions:(C1) Every vector is H-finite and H-continuous, i.e. it is contained ina finite dimensional subspace on which H acts continuously hencesmoothly. (In particular, one can differentiate the H-action and ob-tain an h-action.)(C2) The two actions of the complexified Lie algebra h of H—one throughsubalgebra of g, the other one through differentiation of the H-action,are the same.(C3) The H and g-actions are compatible in the following sense:h · (X · v) = Adh(X) · (h · v)for all h ∈ H,X ∈ g, v ∈ V .If H is compact, then (C1) is equivalent to(C1’) As a H-representation, V is a direct sum of irreducible finite dimen-sional continuous H-representations.The Underlying (g,K)-ModuleWe are particularly interested in the case when H = K is a maximalcompact subgroup. If (pi, V ) is a continuous representation, then the inter-section V sm∩V K of smooth vectors and K-finite vectors is a dense subspaceof V , and stable under g and K-actions, and this intersection satisfies theabove conditions (C1)(C2)(C3).Definition 2.25 ([17] p393). The subspace V sm ∩ V K of a continuous rep-resentation (pi, V ) is called the underlying (g,K)-module of (pi, V ). Con-versely, let V be a (g,K)-module. A continuous representation (pi∗, V∗) of Gis called a G-extension of V if V sm∗ ∩ V K∗ is isomorphic to V as g-moduleand K-representation.Two representations are called infinitesimally equivalent, if their un-derlying (g,K)-modules are isomorphic as (g,K)-modules.42Definition 2.26 ([17] p393). Let V be a (g,K)-module, it is called aHarish-Chandra module, if it satisfies the following conditions:(H1) Any irreducible K-representation occurs in V with finite multiplicity.(Also called admissible in many literatures.)(H2) V is annihilated by some ideal of finite codimension in Z(g).(H3) V is finitely generated over U(g).(H4) V has finite length.(The (H2)(H3)(H4) are equivalent by [17] p408 Lemma 5.3.)2.3.3 Harish-Chandra RepresentationsDefinition 2.27 ([17] p394). A continuous representation (pi, V ) of G iscalled a Harish-Chandra representation, if it satisfies the following con-ditions:• (pi, V ) is a smooth representation, i.e. V sm = V .• (pi, V ) is of moderate growth, in particular it is a Fre´chet representa-tion.• Its (g,K)-module, i.e. V sm ∩ V K is a Harish-Chandra module.Remark 2.28. In [17], it is shown that every finitely generated Harish-Chandra module has a “globalization”, i.e. a Harish-Chandra representa-tion on a nuclear space which has its associated Harish-Chandra module isisomorphic to the given Harish-Chandra module. Therefore, up to infinitesi-mal equivalence, we will only study nuclear Harish-Chandra representationsin this thesis.We will see the following two families of representations are Harish-Chandra representations with nuclear representation space:• Finite dimensional continuous representations;• Smooth parabolic inductions of nuclear Harish-Chandra representa-tions (see the next subsection).2.3.4 Smooth InductionsIn this subsection, we introduce the main object–the smooth parabolicinductions of Harish-Chandra representations. Let G be a real point groupof connected reductive linear algebraic group defined over R, P be a standardreal parabolic subgroup of G.43The Space C∞IndGPσLet (σ, V ) be a smooth Fre´chet representation of the subgroup P , wedefine the following space:C∞IndGPσ := {f ∈ C∞(G,V )| f(pg) = σ(p)f(g), ∀p ∈ P, g ∈ G} (2.8)i.e. the space of smooth functions on G taking values in V , and satisfyingthe “σ-rule”. The G acts on this space by the right regular action: ∀g ∈G, f ∈ C∞IndGPσRg(f)(g′) = (g · f)(g′) := f(g′g), ∀g′ ∈ G. (2.9)The Topology on C∞IndGPσThe elements in the real Lie algebra g0 act as smooth left invariant vectorfields on G. For each f ∈ C∞(G,V ) and X ∈ g0, we define the RXf to be(RXf)(g) := limt→0f(g exp(tX))− f(g)t=ddt|t=0f(g exp(tX)). (2.10)The RXf is still in C∞(G,V ). Moreover, if f is in the subspace C∞IndGPσ,then RXf also satisfies the σ-rule, hence RXf ∈ C∞IndGPσ. By extension ofscalar, we have the g-action hence the algebra action of U(g) on C∞IndGPσ,and we use the same notation RX for all X ∈ U(g). In other words, theC∞IndGPσ is a g-module.Suppose the topology on V is given by a family of semi-norms {| · |ρ : ρ ∈P}. For each element X ∈ g and seminorm ρ on V , we define the followingseminorm on C∞IndGPσ:||f ||X,ρ := supk∈K|RXf(k)|ρ.These semi-norms give the C∞IndGPσ a locally convex Hausdorff topology.The Smooth InductionsUnder the above topology, the right regular G-action on C∞(G,V, σ) isa smooth representation. And we have:Lemma 2.29 ([17] p402 Proposition 4.1). If the (σ, V ) is of moderategrowth, then so is the C∞IndGPσ (under right regular G-action). If (σ, V ) isa Harish-Chandra representation, then so is C∞IndGPσ.Definition 2.30. The C∞IndGPσ with right regular G-action is called thesmooth parabolic induction of (σ, V ) from P to G.442.3.5 Frobenius ReciprocityWe formulate the Frobenius reciprocity (Lemma 2.31), in terms of thesmooth inductions defined in the last subsection. Let G,P be as in thelast subsection, and let (σ, V ) be a Harish-Chandra representation of P andC∞IndGPσ be the smooth parabolic induction.Let e ∈ G be the identity and letΩe : C∞IndGPσ → Vf 7→ f(e)be the delta function at e (evaluation at e). Then Ωe is a continuous linearmap from C∞IndGPσ to V , and it is P -equivariant:Ωe(Rpf) = σ(p)Ωe(f),hence Ωe ∈ HomP (C∞IndGPσ, V ).Lemma 2.31 (Frobenius reciprocity, [17] Lemma 4.2). Let C∞IndGPσbe the smooth induction of (σ, V ) from P to G, let (τ, U) be a smooth rep-resentation of G. Then the following map is a linear isomorphism:HomG(U,C∞IndGPσ)∼−→ HomP (U, V ) (2.11)T 7→ Ωe ◦ TRemark 2.32. It is easy to write down the inverse map of (2.11). For aΦ ∈ HomP (U, V ), we let TΦ : U → C∞IndGPσ be the map defined as follows:TΦ(u)(g) := Φ(g · u), ∀u ∈ U, g ∈ G.It is easy to verify that TΦ(u) ∈ C∞IndGPσ, TΦ ∈ HomG(U,C∞IndGPσ) andΦ 7→ TΦ is the inverse of (2.11).2.3.6 (Hilbert) Normalized Parabolic InductionsIn most references, people realized the parabolic inductions on a Hilbertspace, called normalized parabolic inductions.Let G,P be as above, and let (τ, V ) be a representation of P on a Hilbertspace V . Then let Hτ be the space of smooth functions f ∈ C∞(G,V )satisfyingf(pg) = τ(p)δ1/2P (p)f(g),∀p ∈ P, g ∈ G,45and we define a pre-Hilbert inner product on Hτ by〈f1, f2〉 :=∫K〈f1(k), f2(k)〉τdkwhere 〈, 〉τ is the Hilbert inner product on the Hilbert space V . Then letIndGP τ be the completion of Hτ with respect to the above pre-Hilbert innerproduct. Then right regular G-action on Hτ extends to the entire IndGP τand makes it a Hilbert representation of G.Definition 2.33. The IndGP τ with the right regular G-action, is called thenormalized (Hilbert) parabolic induction of τ (from P to G).Remark 2.34. If (τ, V ) is a unitary representation, the IndGP τ is also aunitary representation. This is the reason to add the factor δ1/2P in thedefinition.There are different ways to define the normalized Hilbert parabolic in-ductions in various of references, and they are all infinitesimally equivalentto the above definition.The relation between the normalized Hilbert parabolic induction andsmooth parabolic induction is as follows:Lemma 2.35. Let τ be a Hilbert representation of P , and let σ = τ ⊗ δ1/2P .Then the C∞IndGPσ and IndGP τ are infinitesimally equivalent.Remark 2.36. The C∞IndGPσ and IndGP τ are both admissible, hence theirG-invariant closed subspaces are in one-to-one correspondence with the(g,K)-submodules of their (isomorphic) underlying (g,K)-modules. There-fore, the C∞IndGPσ and IndGP τ are irreducible/reducible simultaneously.46Chapter 3PreliminarySummary of This ChapterThis assorted chapter lays the foundations for the following chapters,especially for the Chapter 4.The first part consists of section 3.1, and 3.2, in which we introducecertain necessary geometric notions, for the definition of Schwartz functionsin the next chapter. To define the Schwartz functions, one need a scale tomeasure if the functions decrease “rapidly”. As for the classical Schwartzfunctions on Rn, the rapidity of decreasing is measured by derivatives withpolynomial coefficients. However on general smooth manifolds, there is nointrinsic “algebraic functions”. If we look at the special class of manifolds–the Nash manifolds (or nonsingular real algebraic varieties), we do have theintrinsic notion of Nash functions (or polynomial functions), which gives usa scale to define the Schwartz functions on such manifolds.Section 3.3 and 3.4 could be read independently.• Section 3.1 provides a minimal set of knowledge of (affine) real al-gebraic variety. All terms and results are quoted from Chapter 3 of[6].• In 3.2, we review the notion of affine Nash manifolds, following [1] and[2].• In 3.3, we recall the double coset structures on the group G = G(R).Also pay attention to the term “anti-action”, since we will mainly workwith spaces with right translations.• Section 3.4 is pure algebraic. We recall some basic notions on tor-sion submodules, and study the torsion subspace on tensor productmodules.473.1 Real Algebraic VarietiesWe recall the notion of real algebraic varieties, following the Chapter 3of [6]. The notions in this section are from the subject “real algebraic geom-etry”, not from the ordinary algebraic geometry. “Real algebraic varieties”in this thesis have different meaning from “algebraic varieties defined overR”.3.1.1 Algebraic Subsets of RnLet Rn be the n-dimensional real affine space, and let R[X] (also de-noted by R[X1, . . . , Xn]) be the (real coefficients) polynomial ring of n-variables. We denote a general point in Rn by x, or the coordinate formx = (x1, . . . , xn) if necessary.• An algebraic subset of Rn ([6] p23 Definition 2.1.1), is a subset ofthe formZ(S) := {x ∈ Rn : f(x) = 0,∀f ∈ S}where S ⊂ R[X] is a subset.• For a subset Y ⊂ Rn, we denote byI(Y ) := {f ∈ R[X] : f ≡ 0 on Y }the set of polynomials vanishing on entire Y . This is an ideal of R[X],called the vanishing ideal of Y .• The Zariski topology on Rn is the topology whose closed subsetsare algebraic subsets of Rn. On an algebraic subset V ⊂ Rn, theinduced topology from the Zariski topology on Rn is called the Zariskitopology on V . The Euclidean topology on Rn and algebraic subsetV ⊂ Rn are finer than the Zariski topologies on them. The Zariskitopology is not Hausdorff, but is still Noetherian and in particularquasi-compact (every open cover has a finite subcover).The following fact is one of the main differences between real algebraicsubsets and general algebraic sets on arbitrary fields:Lemma 3.1 ([6] p24 Proposition 2.1.3). Every real algebraic subset of Rnis of the form Z(f) for a single polynomial f .483.1.2 Affine Real Algebraic VarietyLet V ⊂ Rn be an algebraic subset.• The quotient ring P(V ) = R[X1, . . . , Xn]/I(V ) is called the ring ofpolynomial functions on V , and elements in it are called polyno-mial functions on V . ([6] p62)• Let U ⊂ V be an Zariski open subset of V . A regular function onU , is a real valued function on U , which could be written as a quotientf/g, for some f, g ∈ P(V ), such that g is nonzero at every point ofU . The regular functions on U form a ring denoted by R(U). ([6] p62Definition 3.2.1)• The correspondence RV : U → R(U) is a sheaf of rings over theZariski topology on V , called the sheaf of regular functions. ([6]p62 Corollary 3.2.4)• An affine real algebraic variety is a pair (X,RX), consisting ofa topological space X isomorphic to an algebraic subset V with itsZariski topology, and a sheaf of rings RX isomorphic to the sheaf ofregular functions RV . ([6] p63 Definition 3.2.9)Zariski closed subsets of an affine real algebraic variety is still an affinereal algebraic variety. By the Lemma 3.1, a Zariski open subset of an affinereal algebraic variety is also an affine real algebraic variety:Lemma 3.2 ([6] p63 Proposition 3.2.10). Let (V,RV ) be an affine realalgebraic variety. Let U ⊂ V be a Zariski open subset. Then the (U,RV |U )is an affine real algebraic variety.This Lemma tells us any Zariski locally closed subset of an affine realalgebraic variety has a natural structure of affine real algebraic variety.Remark 3.3. We can continue to define general real algebraic varietiesby gluing affine real algebraic varieties ([6] p64 Definition 3.2.11). But weprefer to stop here to save space, since all real algebraic varieties studied inthis thesis are locally closed hence affine real algebraic varieties.The affine real algebraic varieties form a special class of affine Nashmanifolds, which will be introduced in the next section.493.2 Basic Notions on Nash ManifoldsThis section is a short introduction to some basic notions in real algebraicgeometry.In the thesis (especially the next chapter), we will frequently quote re-sults from [1] and [2], which are built on Nash manifolds. Thus to make thethesis self-contained but not too long, we write this short section.The real algebraic geometry is not a branch of algebraic geometry, sincethe objects under study are not pure algebraic. However from this sec-tion the reader could see that the pattern to define terms in real algebraicgeometry is similar to that of algebraic geometry.3.2.1 Semi-algebraic Sets and MapsFor each positive integer n, let Rn be the n-dimensional real affine space,and we endow it with the Euclidean topology and canonical smooth struc-ture. Let x1, . . . , xn be the coordinates on Rn, and R[x1, . . . , xn] be the ringof real polynomials on Rn.Definition 3.4. A subset X ⊂ Rn is called a semi-algebraic subset ofRn, if it is a finite union of subsets of the form{x ∈ Rn : fi(x) > 0, gj(x) = 0},for some polynomials fi, gj ∈ R[x1, . . . , xn], i = 1, . . . , r, j = 1, . . . , s.Let X ⊂ Rm, Y ⊂ Rn be two semi-algebraic subsets. A map f : X → Yis called a semi-algebraic map if its graph {(x, f(x)) : x ∈ X} ⊂ Rm ×Rn = Rm+n is a semi-algebraic subset in Rm+n.The finite unions, finite intersections and complements of semi-algebraicsubsets are semi-algebraic. Images of semi-algebraic subsets under semi-algebraic maps are semi-algebraic. The Euclidean closures of semi-algebraicsubsets are semi-algebraic.Example 3.5. We give some examples of semi-algebraic subsets.1. In R, for all −∞ ≤ a < b ≤ ∞, the intervals (a, b), [a, b], [a, b), (a, b]are semi-algebraic subsets in R. Actually all semi-algebraic subsets ofR are (finite) unions of such intervals.2. An algebraic subset of Rn (see 3.1) is semi-algebraic. For example the“cross” in real plane:{(x, y) : x2 − y2 = 0} ⊂ R2.503. Here are some more examples in R2: the quadrant {(x, y) : x > 0, y >0}; the unit cube {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}; a branch of hyperbola{(x, 1/x) : x > 0}.4. The (algebraic) Lie group GL(n,R) = {g ∈Mn(R) : det g 6= 0}, andits connected component GL(n,R)+ = {g ∈ Mn(R) : det g > 0}. Ingeneral, for an algebraic group G defined over R, its real point groupG(R) or a finite union of its Lie group components, are semi-algebraic.Example 3.6. We give some examples of semi-algebraic maps.1. Polynomials in R[x1, . . . , xn] are semi-algebraic maps from Rn to R.2. The absolute value function | · | : R → R, x 7→ |x| is semi-algebraic.The rational power function R→ R, x 7→ xm/n is semi-algebraic (whenit is well-defined).3. Let f1, . . . , fk be semi-algebraic maps from Rn to R, then the functionsmax{f1, . . . , fk} and min{f1, . . . , fk} are semi-algebraic maps.3.2.2 Nash Submanifolds of RnDefinition 3.7 ([2] Definition 2.3.3 and 2.3.5). A semi-algebraic subsetX ⊂ Rn is called a Nash submanifold of Rn, if it is also a smooth regularsubmanifold of Rn.Let X ⊂ Rm be a Nash submanifold of Rn and Y ⊂ Rm be a Nashsubmanifold of Rn. A map f : X → Y is called a Nash map if it is smoothand semi-algebraic.Example 3.8. The semi-algebraic subset {(x, y) : x2 − y2 = 0} is not aregular submanifold of R2 (the origin is a singular point), hence it is not aNash submanifold of R2.Another example is the closed quadrant {(x, y) : x ≥ 0, y ≥ 0}. It is asemi-algebraic subset of R2, but not a regular submanifold (the origin is acorner), hence not a Nash submanifold of R2.Remark 3.9 ([2] Example 2.3.9). Let M ⊂ Rm be a Nash submanifold ofRm, then it has the induced topology from the Euclidean topology on Rm.LetSM := {U ⊂M : U is open in M and semi-algebraic in Rm}be the collection of subsets of M that is (Euclidean) open in M and semi-algebraic in Rm. Then SM is a sub-family of the induced topology on M ,and we have51• The ∅ and M are in SM .• SM is closed under finite union.• SM is closed under finite intersection.Remark 3.10 ([2] Definition 2.3.1 and Example 2.3.9). Still let M ⊂ Rmbe a Nash submaniofld of Rm, and let SM be the collection in the aboveremark. Let U ∈ SM , then U is open in M and semi-algebraic in Rm, henceit is a regular submanifold of Rm and thus also a Nash submanifold of Rm.A real valued function f : U → R is called a Nash function on U , if it isa Nash map in the sense of Definition 3.7. LetNM (U) = {Nash functions on U}be the set of Nash functions on U . Then it is a R-algebra. Conventionallywe let NM (∅) = {0}. For two subsets U1, U2 ∈ SM such that U1 ⊂ U2, onecan restrict a Nash function from U2 to U1 and the restricted function isalso a Nash function on U1.3.2.3 Restricted Topology, Sheaf and CosheafWe recall the notions of restricted topology, sheaves and cosheaves onthem. These notions are introduced in [1] §3.2, appendix A.4 and [2] §2.2.Definition 3.11 ([2] Definition 2.2.1). Let M be a set. A family S ofsubsets of M is called a restricted topology on M , if S contains theempty set and M , and is closed under finite unions and finite intersections.A subset in S is called a restricted open subset ofM , and the complementsof restricted open subsets are called restricted closed subsets of M .A restricted topological space is a pair (M,S) consisting of a setM and a restricted topology S on it. By abuse of terms, we also callM a restricted topological space if there is no ambiguity on the restrictedtopology.A map between two restricted topological spaces is called restrictedcontinuous if every restricted open subset has restricted open preimage.A restricted topology is not a topology, since one cannot take infiniteunions on it.Example 3.12. Let M ⊂ Rm be a Nash submanifold of Rm. Then thefamily SM in Remark 3.9 consisting of semi-algebraic and Euclidean opensubsets of M , is a restricted topology.52Let (M,S) be a restricted topological space. The S is a partially orderedset hence is also a category, one can define the sheaves and cosheaves onrestricted topologies as for sheaves/cosheaves on ordinary topological spaces.The definition is very similar to that of the ordinary sheaves/cosheaves,except that one can only take finite (restricted) open covers.Definition 3.13 ([2] Definition 2.2.4 and 2.2.5). Let C be an abelian cate-gory (e.g. the category of abelian groups/vector spaces/TVS). A presheafon M (with value in C) is a contravariant functor from S to C.More precisely, let F be a presheaf on M , then• For each U ∈ S, one has an object F(U) ∈ C, and in particularF(∅) = {0}.• For each pair of U1, U2 ∈ S with U1 ⊂ U2, one has a restrictionmap resU2U1 : F(U2)→ F(U1) which is a morphism in C. And for threerestricted open subsets U1, U2, U3 ∈ S with U1 ⊂ U2 ⊂ U3 one hasresU3U1 = resU2U1◦ resU3U2 .An element f ∈ F(U) is called a section on U . A morphism between twopresheaves onM is a natural transformation between these two contravariantfunctors.A presheaf F on M (with value in C) is called a sheaf, if it satisfies thefollowing local conditions:• Let U ∈ S be a restricted open subset of M , and {Ui}ki=1 be a finiterestricted open cover of U , i.e. U1, . . . , Uk ∈ S and U = ∪ki=1Ui. Thena f ∈ F(U) is zero if and only if resUUi(f) = 0 for all i = 1, . . . , k.• Let U ∈ S and {Ui}ki=1 be a finite restricted open cover of U . Let fi ∈F(Ui) be a section on Ui for each i. If resUiUi∩Uj (fi) = resUjUi∩Uj (fj) forall 1 ≤ i, j ≤ k, then there exists a f ∈ F(U) such that fi = resUUi(f).Definition 3.14 ([2] Definition 2.2.6 and 2.2.7). Let C be an abelian cat-egory (e.g. the category of abelian groups/vector spaces/TVS). A pre-cosheaf on M with value in C is a covariant functor from S to C.More precisely, let F be a pre-cosheaf on M with value in C, then• For each U ∈ S, one has a object F(U) ∈ C. In particular F(∅) = {0}.53• For each pair of restricted open subsets U1, U2 ∈ S with U1 ⊂ U2, onehas an extension map exU2U1 : F(U1)→ F(U2) which is a morphism inthe category C. And for three restricted open subsets U1 ⊂ U2 ⊂ U3,one hasexU3U1 = exU3U2◦ exU2U1 .An element f ∈ F(U) is called a section on U . A morphism between twopresheaves on M is a natural transformation between these two covariantfunctors.A pre-cosheaf F on M (with value in C) is called a cosheaf, if it satisfiesthe following local conditions:• Let U be a restricted open subset of M and {Ui}ki=1 be a finite re-stricted open cover of U . Then for each f ∈ F(U), there exists afi ∈ F(Ui) such that f =∑ki=1 exUUi(fi). In other words, every sectionon U is a sum of extensions of local sections.• Let U be a restricted open subset and {Ui}ki=1 be a finite restrictedopen cover of U . For each i = 1, . . . , k, let fi ∈ F(Ui) be a local section.Then the∑ki=1 exUUi(fi) = 0 in F(U) if and only if there are fij ∈F(Ui∩Uj) for each i, j ∈ {1, . . . , k} such that fi =∑kj=1[exUiUi∩Uj (fij)−exUiUj∩Ui(fji)].Example 3.15. Let M ⊂ Rm be a Nash submanifold and let SM be therestricted topology defined in Remark 3.9. For each U ∈ SM , let NM (U) bethe ring of Nash functions defined in Remark 3.10, then it is easy to checkthe NM : U 7→ NM (U) is a sheaf on the restricted topology SM .Definition 3.16 ([2] Definition 2.2.9). Let (M,S) be a restricted topolog-ical space and let F be a sheaf over S. Let f ∈ F(M) be a global section,and Z ⊂ M be a restricted closed subset. Then f is supported in Z ifresMM−Z(f) = 0, i.e. f restricts to zero on the complement of Z.Note that on restricted topology, one cannot take infinite intersections ofrestricted closed subsets. Hence there is no “closure” in restricted topology,and in general one cannot define “the support” of a section by taking closuresas in ordinary topological spaces.3.2.4 Affine Nash Manifolds and Abstract Nash ManifoldsSimilar to the definition of affine algebraic varieties and general varieties,we need an intrinsic way to define Nash manifolds without embedding them54into Rm. We first have the following definition which is similar to the ringedspaces in algebraic geometry.Definition 3.17 ([2] Definition 2.3.8). An R-space is a pair (M,OM ) con-sists of a restricted topological space M (with a restricted topology SM )and a sheaf of R-algebras over the restricted topology SM , and the sheafOM is a subsheaf of the sheaf RM of real valued functions on M .A morphism between two R-spaces (M,OM ) and (N,ON ) is a pair(f, f ]) consists of a restricted continuous map f : M → N and a sheafmorphism f ] : f∗RN → RM of restricted sheaves, which maps the subsheaff∗ON to OM .Example 3.18. Let M ⊂ Rm be a Nash submanifold of Rm, and SM bethe restricted topology defined in Remark 3.9, and NM be the sheaf of Nashfunctions defined in Remark 3.10. Then the (M,NM ) is an R-space.Definition 3.19 ([2] Definition 2.3.10). An affine Nash manifold is anR-space which is isomorphic (as R-space) to the R-space of a closed Nashsubmanifold of some Rm.Example 3.20. Any real nonsingular R-affine algebraic variety has a nat-ural structure of affine Nash manifold.Definition 3.21 ([2] Definition 2.3.16). A Nash manifold is an R-space(M,OM ) which has a finite cover {Mi}ki=1 by restricted open subsets ofM , such that each R-space (Mi,OM |Mi) is isomorphic (as R-space) to anaffine Nash manifold. A morphism between two Nash manifolds are justmorphisms of R-spaces between them.All Nash manifolds studied in the thesis are affine Nash manifolds, thuswe stop the introduction here.3.3 Bruhat Decompositions3.3.1 Some Terms on Group ActionsLet G be an abstract group, E be a set. Let AutE be the group ofpermutations on E.• A G-action on E, is a group homomorphism σ : G → AutE, i.e.∀g1, g2 ∈ G, σ(g1g2) = σ(g1) ◦ σ(g2).• A G-(anti)action on E, is a group anti-homomorphism σ : G →AutE, i.e. ∀g1, g2 ∈ G, σ(g1g2) = σ(g2) ◦ σ(g1).55Let G be an abstract group, and H be a subgroup of G.• The left H-translation on G is the H-action h 7→ lh, given bylh : g 7→ hg, ∀g ∈ G.The orbit space of the left H-translation on G is denoted by H\G, itis exactly the set of right H-costs in G.• The right H-translation on G is the H-action h 7→ rh, given byrh : g 7→ gh, ∀g ∈ G.The orbit space of the right H-translation on G is denoted by G/H,it is exactly the set of left H-cosets in G.Let H1, H2 be two subgroup of G.• The left H1-translation on G/H2, is given bygH2 7→ hgH2, ∀h ∈ H1, gH2 ∈ G/H2.The orbit space of this action is denoted by H1\(G/H2).• The right H2-translation on H1\G, is given byH1g 7→ H1gh, ∀h ∈ H2, H1g ∈ H1\G.The orbit space of this anti-action is denoted by (H1\G)/H2.• The (H1, H2)-conjugation on G, is the H1 ×H2-action on G givenbyg 7→ h1gh−12 , ∀g ∈ G, h1 ∈ H1, h2 ∈ H2.The orbit space of this conjugation on G is denoted by H1\G/H2, itis exactly the space of (H1, H2)-double cosets in G.The following easy fact is well-known:Lemma 3.22. The following three orbit spaces are in one-to-one correspon-dence:• The orbit space H1\(G/H2) of left H1-translation on G/H2.• The orbit space (H1\G)/H2 of right H2-translation on H1\G.• The orbit space H1\G/H2 of the (H1, H2)-conjugation on G.563.3.2 Bruhat DecompositionLetG = a connected reductive linear algebraic group defined over RS = a fixed maximal R-split torus of GΣ = the relative root system determined by SW = the relative Weyl group of the root system ΣP∅ = a fixed minimal parabolic R-subgroup containing S∆ = the base of Σ determined by P∅S = the set of simple reflections determined by ∆Σ+ = the positive system spanned by ∆Let Θ,Ω be two subsets of ∆, and PΘ,PΩ be the two standard parabolicR-subgroups corresponding to them, and WΘ,WΩ be the two parabolic sub-group of W generated by simple reflections in Θ,Ω respectively. For eachw ∈ W , one can choose and fix a representative in NG(S)(R). By abuse ofnotation, we denote this fixed representative by the same notation w.The Bruhat DecompositionThe (G(R),P∅(R),NG(S)(R), S) is a Tits system, with real point groupsof parabolic R-subgroups as its parabolic subgroups. By the Remark on page22 of [13], we haveLemma 3.23. The (PΘ(R),PΩ(R))-double cosets on G(R) are in one-to-one correspondence with the double quotient WΘ\W/WΩ. Explicitly, thecorrespondence is given byPΘ(R)\G(R)/PΩ(R)↔WΘ\W/WΩPΘ(R)wPΩ(R)↔WΘwWΩ(Note that we have chosen and fixed a representative for each w ∈ W fromNG(S)(R) ⊂ G(R), and by abuse of notation we denote this representativein G(R) by the same w.)The Double Quotient WΘ\W/WΩAmong the elements in a double coset, one can choose a representativewith minimal length:57Lemma 3.24 (Proposition 1.1.13 in [16]). In each (WΘ,WΩ)-double cosetof W , there is a unique element w characterized by the following equivalentconditions:• w has minimal length in the double coset WΘwWΩ.• w has the minimal length in WΘw and wWΩ.• w−1Θ ⊂ Σ+ and wΩ ⊂ Σ+.We call the element w satisfying the above conditions the minimal repre-sentative of the double coset WΘwWΩ. The set of minimal representa-tives in WΘ\W/WΩ is denoted by[WΘ\W/WΩ] = {w ∈W : w−1Θ ⊂ Σ+, wΩ ⊂ Σ+}.Remark 3.25. In general, for a Coexeter group W and two standardparabolic subgroup WΘ,WΩ. There is a unique minimal element and aunique maximal (under the Bruhat order of W ) element in each doublecoset. The minimal element is exactly the element with minimal length asin the above Lemma. (See [19] Theorem 1.2)There are two partial orders on the set WΘ\W/WΩ of double cosets: oneby the Bruhat order on their maximal elements, and another one by theBruhat order on their minimal elements. It is shown in [29] Theorem 1, thatthese two orders on WΘ\W/WΩ coincide.Double Cosets on GLet G be the Lie group G(R), and similarly PΘ = PΘ(R), PΩ = PΩ(R)be the corresponding Lie groups. The PΘ(R)wPΩ(R) is the same doublecoset as PΘwPΩ in G. We can summarize the above discussion asLemma 3.26. The following four sets are in one-to-one correspondence• The set of (PΘ(R),PΩ(R))-double cosets on G(R);• The set of (PΘ, PΩ)-double coset on G;• The double quotient WΘ\W/WΩ;• The set of minimal representatives [WΘ\W/WΩ].583.3.3 Closure Order on Double CosetsLetG = an algebraic group defined over RP,Q = two closed R-subgroups of GIn the following part of this subsection, we assume G has finitely many(P,Q)-double cosets. In this thesis, we will only consider the case when P,Qare R-parabolic subgroups of a reductive linear algebraic group G defined overR, hence it is safe to make this assumption.The left P-translation and right Q-translation on G are algebraic ac-tions/antiactions defined over R, so is the (P,Q)-conjugation on G. The(P,Q)-double cosets on G are exactly the orbits under the such conjuga-tions. By the Proposition in [10] (p53 §1.8), the double cosets are smoothR-subvarieties of G, and they are open in their closures. The boundaryof each double coset is a disjoint union of double cosets of strictly lowerdimension.Since all the groups and actions are defined over R, one has the similarresults on the real points. Let G,P,Q be the Lie groups of real points ofG,P,Q respectively. The real points of a (P,Q)-double coset is exactly thecorresponding (P,Q)-double coset on G. Under the Zariski topology on G,a (P,Q)-double coset on G is also a real smooth subvariety (in the classicalsense), open in its Zariski closure, with its boundary a union of strictly lowerdimensions. Hence if two double cosets have the same Zariski closure, thenthey are equal.The Closure OrderSuppose the (P,Q)-double cosets in G are parameterized by a finite setI. Let Gi be the double coset corresponding to i ∈ I. Then we have thedisjoint unionG =∐i∈IGi.We define the relation ≤ on the set of double cosets byGi ≤ Gj if and only if Gi ⊂ Gjfor two i, j ∈ I. Here Gj means the Zariski closure of Gj . The bijectionI ↔ {Gi : i ∈ I} thus induces a relation on I.59Lemma 3.27. The relation “≤” is a partial order on {Gi : i ∈ I} (and alsoon I).Proof. First obviouslyGi ⊂ Gi, hence i ≤ i. Second ifGi ≤ Gj andGj ≤ Gi,then Gi = Gj , hence Gi = Gj and i = j. Third if Gi ≤ Gj ≤ Gk, for somei, j, k ∈ I, then Gi ⊂ Gj ⊂ Gk, hence Gi ≤ Gk.Definition 3.28 (Closure Order). We call the above partial order ≤ theclosure order on the set of (P,Q)-double cosets. We denote by Gi < Gj(or i < j) if Gi ≤ Gj but Gi 6= Gj (i ≤ j, i 6= j).Open Unions of Double CosetsFor an i ∈ I, letG≥i :=∐i≤jGj (3.1)G>i :=∐i<jGj = G≥i −GiLemma 3.29. The G≥i and G>i are open in G under the Zariski topology.The Gi is closed in G≥i.Proof. (1) We show G≥i is open by showing the following claim:Claim: For i ∈ I, one hasG≥i = (∪kiGk)c.(“⊃”) First note the complementGc≥i = ∪kiGk ⊂ ∪kiGk,henceG≥i ⊃ (∪kiGk)c.(“⊂”) Second if k i, then Gk ∩ G≥i = ∅. Otherwise, we let Gj ⊂G≥i ∩ Gk, then k ≥ j ≥ i hence k ≥ i a contradiction. Hence Gkc ⊃ G≥i.By taking the intersection over all k i, one hasG≥i ⊂ ∩ki(Gk)c = (∪kiGk)c.By the claim, the G≥i is a complement of a union of closed subsets, henceis open.60(2) We show Gi is closed in G≥i, thus its complement G>i in G≥i is openin G≥i, and also open in G. We show the following claim:Claim: For i ∈ I, one hasGi ∩G≥i = Gi ∩G≥i = Gi.The second equality is trivial. The Gi ∩ G≥i ⊃ Gi ∩ G≥i is also trivial.We just need to show Gi ∩G≥i ⊂ Gi ∩G≥i. Let Gj ⊂ Gi ∩G≥i, then j ≥ iand i ≥ j, hence i = j. Therefore the only double coset in Gi ∩ G≥i is Giitself, henceGi ∩G≥i = Gi = Gi ∩G≥i.Some NotationsLet P = PΘ, and PΩ be two standard parabolic R-subgroups of G. Weare particularly interested in the case whenΩ ⊂ Θ.Let P = PΘ, PΩ be the corresponding groups of real points. In this case, thePΩ is a subgroup of PΘ. The (PΘ, PΩ)-double cosets on G are parameterizedby the minimal representative set [WΘ\W/WΩ], and ordered by the closureorder under Zariski topology.For each w ∈ [WΘ\W/WΩ], we denote byGΩw = the double coset PwPΩGΩ≥w = the Zariski open union of GΩxsuch that GΩx ⊃ GΩw=∐x≥wGΩxGΩ>w = the Zariski open union of GΩxsuch that GΩx ⊃ GΩw and GΩx 6= GΩw=∐x≥w,x6=wGΩxFor the particular case when Ω = ∅ (empty set), we omit the superscript,61and use the following simplified notations:Gw = G∅w = PwP∅G≥w = G∅≥wG>w = G∅>wfor all w ∈ [WΘ\W/W∅] = [WΘ\W ].3.4 Algebraic PreliminaryThis is a pure algebraic section. We study the torsion submodules ontensor product modules over Lie algebras. The main result is Lemma 3.52.In this section, let h be a complex Lie algebra and U(h) be its envelopingalgebra. Let (hk) be the two-sided ideal of U(h) generated by k-products ofelements in h.• In 3.4.1, for a left h-moduleM, we recall the definition of the left anni-hilators M[hk] of the ideal (hk) for each k, and the torsion submoduleM[h•]. The annihilator sequence {M[hk] : k ≥ 0} has no gap and is astrictly ascending sequence (see Lemma 3.38).• In 3.4.2, we define the similar notions as in 3.4.1, for right h-modules.• In 3.4.3, we show a product formula (3.2) on the tensor product moduleM1 ⊗M2 of two left h-modules M1,M2.• In 3.4.4, for two left h-modules M1,M2, we show the tensor productof their torsion submodules is included in the torsion submodule oftheir tensor product (Lemma 3.43):M[h•]1 ⊗M[h•]2 ⊂ (M1 ⊗M2)[h•].In general, the reversed inclusion is not true, see the example at theend of this subsection.• In 3.4.5, we study a h-torsion module M, i.e. M =M[h•]. We definethe depth functiondep :M→ Z≥0of elements in M, and let dep(M) be its image. We show some basicproperties of dep in Lemma 3.46, and show the dep(M) is either Z≥0or a finite consecutive subset of the form {0, 1, . . . , N} in Lemma 3.47.62• In 3.4.6, we study a finite dimensional h-torsion moduleM. We showM has a good basis (3.50), i.e. the annihilator M[hk] is spanned byvectors with depth ≤ k for all k ∈ dep(M).• In 3.4.7, we assume M1 is a finite dimensional h-torsion module, andM2 is an arbitrary left h-module. We show the main theorem of thissection (Lemma 3.52):(M1 ⊗M2)[h•] =M[h•]1 ⊗M[h•]2 =M1 ⊗M[h•]2 .3.4.1 Left Torsion SubmoduleIn this subsection, leth = a complex Lie algebraU(h) = the enveloping algebra of hM = a left h-module(equivalently a left U(h)-module)Definition 3.30. For k ∈ Z>0, let(hk) : = hkU(h)= {X1 · . . . ·Xk · u : ∀X1, . . . , Xk ∈ h,∀u ∈ U(h)}be the right ideal of U(h) generated by k-products of elements in h. Con-ventionally, we let (h0) = U(h).Lemma 3.31. The (hk) is a two-sided ideal for each k ∈ Z>0, and one hasthe following inclusions:(h0) ⊃ (h1) ⊃ (h2) ⊃ . . . ⊃ (hk) ⊃ (hk+1) ⊃ . . . .Definition 3.32. For each k > 0, letM[hk] : = AnnM((hk))= {m ∈M : (hk) ·m = 0}= {m ∈M : X1 · . . . ·Xk ·m = 0,∀X1, . . . , Xk ∈ h}be the left annihilator of the ideal (hk) in M. By convention, we letM[h0] = {0}.63Lemma 3.33. The M[hk] is a left U(h)-submodule of M and one has theinclusions:{0} =M[h0] ⊂M[h1] ⊂M[h2] ⊂ . . . ⊂M[hk] ⊂M[hk+1] ⊂ . . .M.One also has the following easy fact:(hl) · M[hk] ⊂M[hk−l]for all k ≥ l.Definition 3.34. LetM[h•] :=⋃k∈Z≥0M[hk]be the union of all the left annihilatorsM[hk]. We call this the left h-torsionsubspace of M. The module M is called left h-torsion if M[h•] =M.Lemma 3.35. The M[h•] is a left h-submodule of M since it is a directlimit of h-submodules. The annihilators {M[hk] : k ≥ 0} form an exhaustivefiltration of M[h•].Lemma 3.36. The correspondence M 7→M[h•] is a functor from the cate-gory of h-modules to itself.This functor is left exact, i.e. ifM1,M2 are two left h-modules, with aninjective homomorphism of h-modulesM1 ↪→M2, then this homomorphismrestricts to a h-homomorphismM[h•]1 ↪→M[h•]2 ,and we haveM[h•]1 =M1 ∩M[h•]2Remark 3.37. Note that M[h1] is just H0(h,M) = {m ∈ M : X · m =0,∀X ∈ h}.What happens if M[h1] = {0}? Assume this, and let ∀m ∈ M[h2], i.e.X1X2 ·m = 0 for all X1, X2 ∈ h. We fix X2 and let X1 run through h, thenwe see X2 ·m ∈M[h1] = {0}, i.e. X2 ·m = 0. Now let X2 run through h, wesee m ∈ M[h1] = {0}. In sum, M[h2] =M[h1] =M[h0] = {0}. By iteration,we have all annihilators M[hk] = {0}.This works for all k, and we have the following lemma.64Lemma 3.38. LetM be a left h-module, and letM[hk] be the left annihilatorof (hk) for all k ∈ Z≥0. If there is a k ≥ 0, such that M[hk] =M[hk+1], thenM[hk] =M[hk+1] =M[hk+2] = . . .i.e. the annihilator sequence stops ascending.Proof. Suppose M[hk] = M[hk+1] for some k ≥ 0. Let m ∈ M[hk+2], then∀X1, . . . , Xk+1, Xk+2 ∈ h, we have X1 · · ·Xk+1 ·Xk+2 ·m = 0. Hence Xk+2 ·m ∈ M[hk+1]. By the assumption, we also have Xk+2 · m ∈ M[hk]. SinceXk+2 is arbitrary, we see m ∈ M[hk+1]. Hence M[hk+2] = M[hk+1] = M[hk].By iteration, all the following annihilators equal to M[hk].Remark 3.39. Let M be a h-torsion module, then there are only twopossibilities:1. The annihilator sequence is an infinite strictly ascending sequence:{0} =M[h0] $M[h1] $M[h2] $ . . . $M[hk] $M[hk+1] $ . . . ,and all annihilators M[hk] $M are proper subspaces.2. The annihilator sequence is a finite sequence “without jump”, and thereis an N ≥ 0 such that{0} =M[h0] $M[h1] $ . . . $M[hN ] =M,and M =M[hN ] =M[hN+1] =M[hN+2] = . . . .3.4.2 Right Torsion SubmoduleSimilar to the left torsion submodule, we have the right torsion submod-ule of a right h-module. LetM = a right h-module.We still let (hk) be the ideal defined in Definition 3.30.Definition 3.40. For each k ∈ Z>0, letM[hk] = AnnM((hk))= {m ∈M|m · (hk) = 0}= {m ∈M|m ·X1 · . . . ·Xk = 0, ∀X1, . . . , Xk ∈ h}65be the right annihilator of the ideal (hk) in M. Conventionally, we haveM[h0] = {0}.LetM[h•] =⋃k∈Z≥0M[hk]be the union of all right annihilators. We call this union the right h-torsionsubspace of M. We say M is right h-torsion, if M[h•] = M.Similar to the left picture, we haveLemma 3.41. Let M be a right h-module.1. We have the following inclusions:{0} = M[h0] ⊂M[h1] ⊂ . . .M[hk] ⊂M[hk+1] ⊂ . . . ⊂M,and each M[hk] is a right h-submodule of M.2. For k ≥ l, we haveM[hk] · (hl) ⊂M[hk−l].3. The {M[hk] : k ≥ 0} form an exhaustive filtration on M[h•], and M[h•]is a right h-submodule of M.3.4.3 Tensor Product ModuleLet M1,M2 be two left h-modules, and let M1 ⊗M2 be their tensorproduct module. In this subsection we show a multiplication formula (3.2)on the tensor product module M1 ⊗M2.Let m1 ∈M1,m2 ∈M2 be arbitrary elements. Then an element X ∈ hacts on m1 ⊗m2 ∈M1 ⊗M2 asX · (m1 ⊗m2) = X ·m1 ⊗m2 +m1 ⊗X ·m2.For a positive integer k, let[1, k] = {1, 2, . . . , k}be the ordered set of the first kth positive integers. Let X1, . . . , Xk bearbitrary k elements in h. For a subset S ⊂ [1, k], we label the elements inS asS = {i1, i2, . . . , is} with i1 < i2 < . . . < isAnd we use the following notation to denote the ordered product in U(h):XS = Xi1 ·Xi2 · · ·Xis ∈ U(h).By convention, we let X∅ = 1.66Lemma 3.42. For arbitrary m1 ∈M1,m2 ∈M2 and elements X1, . . . , Xkin h, we have the following multiplication formulaX1 ·X2 · · ·Xk · (m1 ⊗m2) =∑S⊂[1,k](XS ·m1)⊗ (XSc ·m2). (3.2)The Sc ⊂ [1, k] is the complement of S in [1, k]. The right hand side is asum of 2k terms when S run through all subsets of [1, k].Proof. We proceed by induction on k. For k = 1, the formula is trivial. The[1, 1] = {1} has only two subsets ∅ and [1, 1] itself. The right hand side of(3.2) is(X∅ ·m1)⊗ (X[1,1] ·m2) + (X[1,1] ·m1)⊗ (X∅ ·m2)which is exactly m1 ⊗X1 ·m2 +X1 ·m1 ⊗m2.Assume the formula is true for k = n− 1. Then for k = n, we haveX1 ·X2 · · ·Xn(m1 ⊗m2)= X1 · (X2 · · ·Xn · (m1 ⊗m2))= X1 · (∑S⊂[2,n](XS ·m1)⊗ (XSc ·m2)) (induction hypothesis)=∑S⊂[2,n](X1 ·XS ·m1)⊗ (XSc ·m2) +∑S⊂[2,n](XS ·m1)⊗ (X1 ·XSc ·m2)=∑S⊂[1,n]1∈S(XS ·m1)⊗ (XSc ·m2) +∑S⊂[1,n]1∈Sc(XS ·m1)⊗ (XSc ·m2)=∑S⊂[1,n](XS ·m1)⊗ (XSc ·m2)3.4.4 Torsion Submodule of Tensor Products—ILetM1,M2 be two left h-modules, andM1⊗M2 be their tensor productmodule. We discuss the relation between their left h-torsion subspaces.Let M[h•]1 ,M[h•]2 and (M1 ⊗M2)[h•] be the left h-torsion subspaces ofM1,M2,M1 ⊗M2 respectively. We first have the following easy fact:Lemma 3.43. The tensor product of torsion subspaces is contained in thetorsion subspace of tensor product:M[h•]1 ⊗M[h•]2 ⊂ (M1 ⊗M2)[h•] (3.3)67i.e. M[h•]1 ⊗M[h•]2 is a left h-submodule of (M1 ⊗M2)[h•]. More precisely,we haveM[hk1 ]1 ⊗M[hk2 ]2 ⊂ (M1 ⊗M2)[hk1+k2 ] (3.4)for all k1, k2 ∈ Z≥0.Proof. The module structures are clear, and we just need to show thestronger inclusion (3.4). Let m1 ∈ M[hk1 ]1 and m2 ∈ M[hk2 ]2 , i.e. m1 iskilled by k1-products of elements in h and m2 is killed by k2-products ofelements in h.Let k = k1 + k2, and let X1, . . . , Xk be arbitrary elements in h. ByLemma 3.42, we have the product formulaX1 ·X2 · . . . ·Xk · (m1 ⊗m2) =∑S⊂[1,k](XS ·m1)⊗ (XSc ·m2).For each S ⊂ [1, k] on the right hand side, either |S| ≥ k1 or |Sc| ≥ k2 istrue. Otherwise we have k = |S| + |Sc| < k1 + k2 = k, a contradiction!Hence for each S ⊂ [1, k], either XS ·m1 = 0 or XSc ·m2 = 0. Hence theright hand side is always zero.Since the X1, . . . , Xk are arbitrary, we see m1 ⊗m2 is in the annihilatorAnnM1⊗M2((hk)) = (M1 ⊗M2)[hk] ⊂ (M1 ⊗M2)[h•].In general, the torsion subspace (M1⊗M2)[h•] is larger than the tensorproduct M[h•]1 ⊗M[h•]2 , see the following example.Example 3.44. Leth = {x : x ∈ C}be the one dimensional abelian Lie algebra. LetV = C = {y : y ∈ C}be a one dimensional h-module, with the h-action given byx · y = xy,i.e. the scalar multiplication on C. Then one can seeV [h1] = {y ∈ C : xy = 0,∀x ∈ C} = {0}and similarlyV [hk] = {y ∈ C : x1x2 . . . xky = 0,∀x1, . . . , xk ∈ C} = {0}.68Hence one hasV [h•] = {0}.LetVˆ = C = {z : z ∈ C}be the dual h-module of V , with the h-action given byx · z = −xz.Then by the same argument one can seeVˆ [hk] = 0,∀k ≥ 0and Vˆ [h•] = {0}.The tensor product V ⊗ Vˆ is a one dimensional trivial h-module, i.e.∀y, z ∈ C, we havex · (y ⊗ z) = xy ⊗ z + y ⊗ (−xz) = 0.Hence(V ⊗ Vˆ )[h•] = (V ⊗ Vˆ )[h1] = V ⊗ Vˆ ,which is larger than V [h•] ⊗ Vˆ [h•] = {0}.3.4.5 Depth and Torsion ModulesLet h be a complex Lie algebra and M be a left h-module. In thissubsection, we assume M is h-torsion, i.e.M =M[h•].The filtration by annihilators{0} =M[h0] ⊂M[h1] ⊂ . . .M[hk] ⊂M[hk+1] ⊂ . . .is an exhaustive filtration of M, i.e. for all m ∈ M, there exists a k suchthat m ∈M[hk].Definition 3.45. For a h-torsion left h-module M, we define the depthfunction on M bydep(m) = min{k : m ∈M[hk]}, ∀m ∈M. (3.5)69Namely, the depth of m ∈ M is the minimal non-negative integer k suchthat m is annihilated by (hk). The dep :M→ Z≥0 is a function onM withvalues in non-negative integers, and we denote its image bydep(M) = {dep(m) : m ∈M}i.e. the set of depth of elements in M.We have the following easy facts:Lemma 3.46. Let m ∈M, then(1) dep(m) = 0 if and only if m = 0.(2) dep(m) = 1 if and only if m ∈ H0(h,M) but m 6= 0.(3) M[hk] = {m ∈M : dep(m) ≤ k}.(4) For k > 0, the {m ∈M : dep(m) = k} =M[hk] −M[hk−1].(5) For arbitrary X ∈ h, we have dep(X · m) < dep(m), i.e. X strictlyreduce the depth.(6) For c ∈ C, we havedep(cm) ={dep(m) if c 6= 00 if c = 0(7) Let m1,m2 ∈M, thendep(m1 +m2) ≤ max{dep(m1),dep(m2)}. (3.6)Proof. (1)(2)(3)(4)(6) are obvious by definition. We show (5) and (7).(5): Let k = dep(m) and X ∈ h be an arbitrary element. By definitionwe have (hk) ·m = {0}. Then(hk−1) · (X ·m) ⊂ (hk) ·m = {0},hence X ·m ∈M[hk−1]. Again by definition of dep, we havedep(X ·m) = min{i : X ·m ∈M[hi]} ≤ k − 1 < k,i.e. dep(X ·m) < k = dep(m).70(7): Let k1 = dep(m1), k2 = dep(m2), k = max{k1, k2}. Thenm1 ∈M[hk1 ] ⊂M[hk]m2 ∈M[hk2 ] ⊂M[hk]Hence m1 +m2 ∈M[hk], and by definitiondep(m1 +m2) ≤ k = max{k1, k2} = max{dep(m1),dep(m2)}.Lemma 3.47. The depth function dep : M → Z≥0 is either onto or withfinite consecutive image. More precisely, the image dep(M) is either Z≥0or of the form {0, 1, . . . , k} form some k ≥ 0.Proof. This follows immediately from Remark 3.39.In sum, the dep(M) is a well-ordered set, we can perform inductions onthe depth of the module M.3.4.6 Finite Dimensional Torsion ModulesIn this subsection, we look at the special case whenM is a finite dimen-sional h-torsion module. First we haveLemma 3.48. Let M be a finite dimensional h-torsion module. Then thedepth function dep on M is bounded, hence dep(M) is a finite consecutivesubset of Z≥0.Proof. This follows immediately from Lemma 3.47.Lemma 3.49. Let M be a finite dimensional h-torsion module, and letdep(M) = {0, 1, . . . , N},then(1) The annihilator sequence of M is{0} =M[h0] $M[h1] $ . . . $M[hN ] =Mand for all k ≥ N , the M[hk] =M.71(2) For any m ∈ M and k ∈ dep(M), we have dep(m) = k if and only ifm ∈M[hk] −M[hk−1].Proof. Part (1) follows from the above lemma. Part (2) is just a reformula-tion of (4) of Lemma 3.46.Let {m1, . . . ,md} be a basis of M (d = dimM). For each k ∈ dep(M),we consider the subspacespan{mi : 1 ≤ i ≤ d,dep(mi) ≤ k}spanned by all basis vectors with depth less than or equal to k. Obviouslythis space is contained in M[hk] by (6) (7) of Lemma 3.46.Definition 3.50. Let M be a finite dimensional h-torsion module, anddep(M) = {0, 1, . . . , N}. A basis {m1, . . . ,md} of M is called a goodbasis, ifM[hk] = span{mi : 1 ≤ i ≤ d,dep(mi) ≤ k}for all k ∈ dep(M).Lemma 3.51. Let M be a finite dimensional h-torsion module. Then ithas a good basis.Proof. We construct a good basis. Let dep(M) = {0, 1, . . . , N}. Letdk = dimM[hk]be the dimension of the kth annihilator, we have 0 = d0 < d1 < d2 < . . . <dN = dimM (strictly increasing).First let {m1, . . . ,md1} be an arbitrary basis of M[h1]. Since they arebasis vectors, they are non-zero, hence dep(mi) = 1 for all 1 ≤ i ≤ d1.ObviouslyM[h1] = span{m1, . . . ,md1}.SinceM[h1] is a finite dimensional (proper) subspace ofM[h2] (also finitedimensional), one can extend the basis {m1, . . . ,md1} of M[h1] to a basis{m1, . . . ,md1 ,md1+1, . . . ,md2} of M[h2]. Obviously, the d2 − d1 new basisvectors {md1+1, . . . ,md2} are in the complement M[h2] −M[h1], hence alltheir depth equal to 2 by (4) of Lemma 3.46. Obviously we haveM[h2] = span{m1, . . . ,md2}.By iteration, for each k < N , assume we have constructed a basis{m1, . . . ,mdk} of M[hk], such that {m1, . . . ,mdi} is a basis of the subspace72M[hi], and dep(mdi−1+1) = . . . = dep(mdi) = i for all i ≤ k. Then we canextend the basis {m1, . . . ,mdk} of M[hk] to a basis{m1, . . . ,mdk ,mdk+1, . . . ,mdk+1}of M[hk+1]. The dk+1 − dk new vectors {mdk+1, . . . ,mdk+1} are obviously inthe complement M[hk+1] −M[hk] and all their depth equal to k + 1 by (4)of Lemma 3.46.This iteration terminate when k = N , and we then obtain a basis{m1, . . . ,mdN } which is obviously a good basis.3.4.7 Torsion Submodule of Tensor Products—IIIn this subsection, we show the reversed inclusion of Lemma 3.43. Let hbe a complex Lie algebra,M1,M2 be two left h-modules andM1⊗M2 betheir tensor product module.Moreover, we assume M1 is• h-torsion, i.e. M1 =M[h•]1 ;• finite dimensional.We haveLemma 3.52. Let M1 be a finite dimensional h-torsion module, and M2be an arbitrary left h-module. Then(M1 ⊗M2)[h•] =M[h•]1 ⊗M[h•]2 =M1 ⊗M[h•]2 .We have seen one inclusion in Lemma 3.43, we just need to show thereversed inclusion(M1 ⊗M2)[h•] ⊂M[h•]1 ⊗M[h•]2 .Let dep(M1) = {0, 1, . . . , N}. Then the {M[hk]1 : 0 ≤ k ≤ N} is a finitefiltration which is strictly ascending. Let dk = dimM[hk]1 , ∀k ∈ dep(M1)and let {u1, . . . , udN } be a good basis ofM1 (Definition 3.50), and we labelthem with non-decreasing depth:dep(ui) = k, dk−1 + 1 ≤ i ≤ dkfor all k = 1, . . . , N .73The submodules {M[hk]1 ⊗M2 : 0 ≤ k ≤ N} form a finite exhaustivefiltration ofM1⊗M2, and the submodules {(M[hk]1 ⊗M2)[h•] : 0 ≤ k ≤ N}form a finite exhaustive filtration of (M1 ⊗M2)[h•]. We just need to showthe inclusion(M[hk]1 ⊗M2)[h•] ⊂M[h•]1 ⊗M[h•]2 ,or more precisely the inclusions(M[hk]1 ⊗M2)[hl] ⊂M[h]1 ⊗M[h•]2 ,for each l ≥ 0, by (finite) induction (iteration) on k.Proof. For k = 0, M[h0]1 = {0} and (M[h0]1 ⊗M2)[h•] = {0} which is obvi-ously contained in M[h•]1 ⊗M[h•]2 .For k = 1, recall that the M[h1]1 has basis {u1, . . . , ud1}, and everyelement in M[h1]1 ⊗M2 is uniquely written asd1∑i=1ui ⊗ vifor some vi ∈M2, 1 ≤ i ≤ d1. And it is zero if and only if all vi are zero. Ifmoreover the element∑d1i=1 ui ⊗ vi is in(M[h1]1 ⊗M2)[hl]then for any X1, . . . , Xl ∈ h, one hasX1 · · ·Xl · (d1∑i=1ui ⊗ vi) = 0.By Lemma 3.42, one hasX1 · · ·Xl · (d1∑i=1ui ⊗ vi) =d1∑i=1X1 · · ·Xl · (ui ⊗ vi)=d1∑i=1∑S⊂[1,l]XSui ⊗XScvi74In the last term, if S 6= ∅, then XSui = 0 since ui ∈ M[h1]1 . Hence onlyS = ∅ contributes to the last sum, and we haveX1 · · ·Xl · (d1∑i=1ui ⊗ vi) =d1∑i=1ui ⊗X[1,l]vi=d1∑i=1ui ⊗X1 · · ·Xl · viIf this sum is zero, then X1 · · ·Xl · vi = 0 for all 1 ≤ i ≤ d1. Thus vi ∈M[h•]2since the X1, . . . , Xl are arbitrary. Hence the original elementd1∑i=1ui ⊗ vi ∈M[h1]1 ⊗M[hl]2 .We thus have shown(M[h1]1 ⊗M2)[hl] ⊂M[h1]1 ⊗M[hl]2 ⊂M[h•]1 ⊗M[h•]2 ,thus we have(M[h1]1 ⊗M2)[h•] ⊂M[h•]1 ⊗M[h•]2 .(Induction Hypothesis) Assume(M[hk−1]1 ⊗M2)[h•] ⊂M[h•]1 ⊗M[h•]2 .We need to show (M[hk]1 ⊗M2)[hl] ⊂M[h•]1 ⊗M[h•]2 , for all l ≥ 0.For general k, a generic element in M[hk]1 ⊗M2 is uniquely written asdk∑i=1ui ⊗ vifor some vi ∈M2. If moreover it is in (M[hk]1 ⊗M2)[hl], thenX1 · · ·Xl · (dk∑i=1ui ⊗ vi) = 075for all X1, . . . , Xl ∈ h. By Lemma 3.42, we haveX1 · · ·Xl · (dk∑i=1ui ⊗ vi) =dk∑i=1X1 · · ·Xl · (ui ⊗ vi)=dk∑i=1∑S⊂[1,l]XSui ⊗XScvi=∑S⊂[1,l]dk∑i=1XSui ⊗XScvi=dk∑i=1ui ⊗X[1,l]vi +∑S⊂[1,l]S 6=∅dk∑i=1XSui ⊗XScvi=dk∑i=dk−1+1ui ⊗X[1,l]vi +dk−1∑i=1ui ⊗X[1,l]vi+∑S⊂[1,l]S 6=∅dk∑i=1XSui ⊗XScviThe second term∑dk−1i=1 ui ⊗X[1,l]vi is in M[hk−1]1 ⊗M2 since ui ∈ M[hk−1]1for all i ≤ dk−1. The third term∑S⊂[1,l]S 6=∅∑dki=1XSui ⊗ XScvi is also inM[hk−1]1 ⊗M2, since all XSui are in M[hk−1]1 . (Remember that ui ∈ M[hk]1and h strictly reduce the depth, hence XSui ∈M[hk−1]1 when S is nonempty.)In sum, the last two terms are all in M[hk−1]1 ⊗M2, and the first term∑dki=dk−1+1 ui⊗X[1,l]vi is linear independent from theM[hk−1]1 ⊗M2. Hencethe X1 · · ·Xl · (∑dki=1 ui ⊗ vi) is zero if and only if bothdk∑i=dk−1+1ui ⊗X[1,l]vi = 0anddk−1∑i=1ui ⊗X[1,l]vi +∑S⊂[1,l]S 6=∅dk∑i=1XSui ⊗XScvi = 0.76The∑dki=dk−1+1 ui⊗X[1,l]vi = 0 implies X[1,l]vi = X1 · · ·Xl ·vi = 0 for alli such that dk−1 +1 ≤ i ≤ dk. Hence the vi ∈M[h•]2 for all dk−1 +1 ≤ i ≤ dk,anddk∑i=dk−1+1ui ⊗ vi ∈M[hk]1 ⊗M[h•]2 . (3.7)Thendk∑i=dk−1+1ui ⊗ vi ∈M[hk]1 ⊗M[h•]2 ⊂ (M[hk]1 ⊗M2)[h•](replacing M1 by M[hk]1 in Lemma 3.43). Hencedk−1∑i=1ui⊗vi =di∑i=1ui⊗vi−dk∑i=dk−1+1ui⊗vi ∈ (M[hk]1 ⊗M2)[h•] ⊂ (M1⊗M2)[h•].Also note that (good basis)dk−1∑i=1ui ⊗ vi ∈M[hk−1]1 ⊗M2.Hencedk−1∑i=1ui ⊗ vi ∈ (M[hk−1]1 ⊗M2) ∩ (M1 ⊗M2)[h•] = (M[hk−1]1 ⊗M2)[h•].By the induction hypothesis, we havedk−1∑i=1ui ⊗ vi ∈M[h•]1 ⊗M[h•]2 (3.8)Combining (3.7) with (3.8), we havedk∑i=1ui ⊗ vi ∈M[h•]1 ⊗M[h•]2 .Hence(M[hk]1 ⊗M2)[hl] ⊂M[h•]1 ⊗M[h•]2for all l ≥ 0, and(M[hk]1 ⊗M2)[h•] ⊂M[h•]1 ⊗M[h•]2 .77Chapter 4Schwartz Functions,Schwartz Inductions andSchwartz DistributionsSummary of This ChapterIn this chapter, we build the toolbox of Schwartz analysis for the study ofintertwining operators (distributions). This chapter consists of three parts,the first part mainly follows the work [1] and [2], the second part is on somefundamental facts, and the third part is innovative and it combines the workof Aizenbud-Gourevitch with the notion of Schwartz inductions introducedin [20].• The first part consists of section 4.1 (some notions), 4.2, 4.3 and 4.4,in which we recall the theory of Schwartz functions and distributionson affine Nash manifolds, developed in [1] and [2]. The key points ofthese four sections are– Schwartz R-valued, C-valued, E-valued functions, (the spacesS(M,R), S(M), S(M,E)).– Crucial properties of Schwartz functions: Lemma 4.21, Proposi-tion 4.30.– Schwartz distributions (Definition 4.31), restrictions of distribu-tions (Definition 4.34) and crucial properties (Lemma 4.32, 4.35,4.36).– The term “supported in”, and the result: the space of Schwartzdistributions supported in a closed Nash submanifold is indepen-dent of neighbourhoods (Lemma 4.38, Lemma 4.39).• The second part is section 4.5, where we apply the notions and resultsin the first part to the nonsingular affine real algebraic varieties or78even real points of algebraic groups, which is a special class of affineNash manifolds. The key points of this section are– The Zariski topology on a nonsingular affine real algebraic vari-ety is included in the restricted topology. Therefore by restrict-ing to Zariski topology, one has the pseudo-cosheaf of Schwartzfunctions on Zariski topology and pseudo-sheaf of Schwartz dis-tributions on Zariski topology.– The Zariski topology is quasi-compact, hence a Schwartz distri-bution on a nonsingular affine real algebraic variety has a well-defined support, which is a Zariski closed subset.– Let G be a real algebraic group, Y ⊂ G be a subvariety stableunder right translation of an algebraic subgroup H. Then theSchwartz function space S(Y,E) is preserved under right regularH-action and is a smooth H-representation.• The third part is section 4.6, in which we combine the notion ofSchwartz inductions in [20] with the above theory of Schwartz func-tions/distributions, to build a distribution theory on the local Schwartzinductions. The key points of this part are– Schwartz inductions SIndGPσ and local Schwartz inductions (Def-inition 4.57, 4.64). The Schwartz induction SIndGPσ is exactly thesmooth induction C∞IndGPσ when the quotient manifold P\G iscompact.– Properties of local Schwartz inductions: NF-space (Lemma 4.65),open extensions (Lemma 4.67), closed restrictions (Lemma 4.69).– Zariski P -topology, SInd−Pσ is a pseudo-cosheaf on Zariski P -topology (Lemma 4.73).– Distributions on Schwartz inductions (Definition 4.76), pseudo-sheaf property (Lemma 4.79), independency of neighbourhoods(Lemma 4.80).– Right regular actions on Schwartz inductions.– Schwartz inductions on fibrations: Lemma 4.91 and Lemma 4.92.794.1 Some Notions4.1.1 Nash Differential OperatorsLet M be an affine Nash manifold, then it has a canonical structure ofsmooth manifold. Let C∞(M,R) be the ring of smooth real valued functionson M .Definition 4.1 ([1] Definition 3.5.1). Let M be an affine Nash manifold, andlet HomR(C∞(M,R), C∞(M,R)) be the ring of R-linear homomorphisms onthe R-algebra C∞(M,R). LetDiffNash(M)be the subring (R-subalgebra) of HomR(C∞(M,R), C∞(M,R)) generatedby the ring N (M) of Nash functions and Nash vector fields. We call it thering of Nash differential operators on M .Remark 4.2. Each restricted open subset U ⊂ M is also an affine Nashmanifold, and one can define the ring DiffNash(U) of Nash differential oper-ators on U . Since Nash functions and Nash vector fields form sheaves on therestricted topology, one can define the natural restriction of a Nash differen-tial operator in DiffNash(M) to U , and obtain a Nash differential operator inDiffNash(U). This restriction makes the correspondence U 7→ DiffNash(U) asheaf on the restricted topology. We call it the sheaf of Nash differentialoperators on M , and denote it by DiffNash : U 7→ DiffNash(U).4.1.2 Affine Real Algebraic Varieties as Affine NashManifoldsLet X be a nonsingular affine real algebraic variety ([6] p67 Definition3.3.9). It has a canonical structure of affine Nash manifold ([1] Example3.3.4). More precisely, we can embed X into an affine space Rn as a realalgebraic set. Under the Euclidean topology on Rn, the X has a canonicalstructure of real analytic space. Since the X is nonsingular as real algebraicvariety, the real analytic space is actually a smooth manifold (and also aclosed regular submanifold of Rn). It is obviously semi-algebraic since it isalgebraic. Hence X has a canonical structure of affine Nash manifold. Thethree structures of affine real algebraic variety, smooth manifold, and affineNash manifold are all intrinsic and independent of the embedding X ⊂ Rn.80LetT Zar = the Zariski topology on XT Res = the restricted topology on XT Euc = the Euclidean topology on XThe T Zar and T Euc are topologies on X, while the T Res is only a restrictedtopology on X (Definition 3.11). We will call subsets in these “topologies”Zariski open, restricted open, Euclidean open subsets respectively. Itis obvious thatT Zar ⊂ T Res ⊂ T Euc.LetRX = the sheaf of regular functions on (X, T Zar)NX = the (restricted) sheaf of Nash functions on (X, T Res)C∞(−,R) = the sheaf of real-valued smooth functions on (X, T Euc)These are sheaves on the Zariski topology, restricted topology and Euclideantopology respectively.Let U ∈ T Zar be a Zariski open subset of X, it is also a restricted opensubset and Euclidean open subset. We have the inclusions of rings:P(U) ⊂ RX(U) ⊂ NX(U) ⊂ C∞(U,R).Here the four rings are rings of polynomial functions on U , regular functionson U , Nash functions on U and smooth functions on U . All these rings arerings of R-valued functions.Similar to the Nash differential operators in Definition 4.1, we haveDefinition 4.3 ([1] Definition 3.5.1). Let X be a nonsingular affine realalgebraic variety, which is also regarded as a smooth manifold and affineNash manifold. Let C∞(X,R) be the ring of real-valued smooth functionson X. LetDiffAlg(X)be the subring of HomR(C∞(M,R), C∞(M,R)) generated by the ring P(X)of polynomial functions and algebraic vector fields. We call this ring thering of algebraic differential operators on X.814.2 Schwartz Functions on Affine Nash Manifolds4.2.1 Definition of Schwartz FunctionsIn this subsection, we recall the notion Schwartz R-valued functions de-fined in [1].Schwartz Functions on Affine Nash ManifoldsLet M be an affine Nash manifold.Definition 4.4 ([1] Definition 4.1.1). A smooth function f ∈ C∞(M,R)is called a Schwartz function on M , if for all Nash differential operatorD ∈ DiffNash(M), the function Df is bounded on M , i.e.supx∈M|(Df)(x)| <∞, ∀D ∈ DiffNash(M).We denote byS(M,R)the space of Schwartz functions on M .For each D ∈ DiffNash(M), we have a seminorm | · |D on S(M,R) givenby|f |D := supx∈M|(Df)(x)|.The seminorms {| · |D : D ∈ DiffNash(M)} define a locally convex Hausdorfftopology on S(M,R) and make it into a topological vector space over R.Lemma 4.5 ([1] Corollary 4.1.2 and [2] Corollary 2.6.2). The space S(M,R)is a nuclear Fre´chet space, under the topology defined by the seminorms{| · |D : D ∈ DiffNash(M)}.Schwartz Functions on Nonsingular Affine Real AlgebraicVarietiesLet X be a nonsingular affine real algebraic variety, then it has a naturalstructure of affine Nash manifold ([1] Example 3.3.4). We can define theS(M,R) of Schwartz functions by regarding it as an affine Nash manifold.However the Schwartz functions could be equivalently defined by algebraicdifferential operators.82Lemma 4.6 ([1] Lemma 3.5.5). Let X be a nonsingular affine real algebraicvariety. The ring DiffNash(X) of Nash differential operators on X is gen-erated by the subring DiffAlg(X) of algebraic differential operators and thesubring N (X) of Nash functions on X.Lemma 4.7 ([1] Proposition 4.1.1 and Corollary 4.1.3). A function f ∈C∞(X,R) is in S(M,R) if and only if Df is bounded on X for all algebraicdifferential operators D ∈ DiffAlg(X). Hence Schwartz functions on X couldbe defined as smooth functions with all algebraic derivatives bounded on X.Remark 4.8. By the above Lemma, for the special case of affine real al-gebraic varieties, the notion of Schwartz functions in [1] is the same as thenotion of Schwartz functions studied in [20].4.2.2 Properties of Schwartz FunctionsWe summarize the crucial properties of Schwartz function spaces, fordetails see [1].Open Extensions of Schwartz FunctionsLet M be an affine Nash manifold, and let U be a restricted open subset(hence also an affine Nash manifold). Let S(M,R),S(U,R) be the corre-sponding spaces of Schwartz functions.Let f ∈ S(U,R) be a Schwartz function on U , we define the naive ex-tension exMU f of f to M by(exMU f)(x) :={f(x), if x ∈ U0, if x ∈M − U (4.1)Lemma 4.9 ([1] Proposition 4.3.1, or [20] p265 Theorem 1.2.4(i)). Thenaive extension exMU f is in S(M,R). The mapS(U,R)→ S(M,R) (4.2)f 7→ exMU fis an injective continuous linear map, called extension by zero from Uto M . Its image is exactly the subspace{f ∈ S(M,R) : Df ≡ 0 on M − U,∀D ∈ DiffNash(M)},which is a closed subspace of S(M,R), hence the above extension map hasclosed image and is a homomorphism of TVS.83Partition of UnityLemma 4.10 ([1] Theorem 4.4.1, or [20] p268 Lemma 1.2.7). Let M bean affine Nash manifold, and let {Ui}ki=1 be a finite family of restrictedopen subsets of M , covering M . Then there exist smooth functions αi ∈C∞(M,R), i = 1, . . . , k such that suppαi ⊂ Ui and∑ki=1 αi ≡ 1 on M .Moreover, we can choose αi such that for any f ∈ S(M,R), the αif ∈S(Ui,R).Remark 4.11. The αi in this Lemma could be chosen to be temperedfunctions ([1] Definition 4.2.1), but we don’t need this notion.Cosheaf PropertyLet U1, U2 be two restricted open subsets of M , and S(U1,R),S(U2,R)be the Schwartz function spaces over them. Assume U1 ⊂ U2, then the U1is a restricted open subset of the affine Nash manifold U2, and we have theextension map exU2U1 : S(U1,R)→ S(U2,R) as in Lemma 4.9. This makes thecorrespondence S(−,R) : U 7→ S(U,R) into a pre-cosheaf on the restrictedtopology on M . Actually we haveLemma 4.12 ([1] Proposition 4.4.4). The S(−,R) : U → S(U,R) is acosheaf on the restricted topology of M .Closed Restrictions of Schwartz FunctionsIn [1], the authors use the term “affine Nash submanifold Z ⊂ M”without defining it. We make the following definition, which is sufficient forour use.Definition 4.13. Let M ⊂ Rn be a Nash submanifold of Rn which isclosed under the Euclidean topology. A subset Z ⊂ M is called a closedsubmanifold of the Nash submanifold M ⊂ Rn, if• Z ⊂ Rn is a Nash submanifold of Rn, and is closed in M under theEuclidean topology.• the embedding Z ↪→ M is a Nash map and regular embedding ofmanifolds.With these two conditions, the inclusions Z ↪→ Rn and Z ↪→M are Nashmaps, Z is closed in M and Rn, and Z is indeed a regular submanifold ofM in the sense of smooth manifolds.84Remark 4.14. In [1], the authors give a definition of general Nash manifoldthrough ringed space as in algebraic geometry. In particular, the notion of“affine Nash manifold” (Definition 3.19 or [1] Definition 3.3.1) is defined as aringed space, without embedding into any Rn. However, we will only studyaffine Nash manifolds, and could find an embedding of them into Euclideanspaces.Lemma 4.15 ([1] Theorem 4.6.1, or [20] p266 Theorem 1.2.4 (iii)). LetM ⊂ Rn be a closed Nash submanifold of Rn, and Z ⊂ M is a closedNash submanifold in the sense of Definition 4.13. For a Schwartz functionf ∈ S(M,R), let f |Z be its restriction to Z. Then f |Z ∈ S(Z,R), and themapS(M,R)→ S(Z,R) (4.3)f 7→ f |Zis a continuous linear map. It is surjective, hence is a homomorphism ofTVS.External Tensor ProductLet M1,M2 be two affine Nash manifolds, and their direct product M1×M2 has the natural structure of affine Nash manifold. We haveLemma 4.16 ([2] Corollary 2.6.3, or [20] p268 Proposition 1.2.6(ii)). Thenatural mapS(M1,R)⊗ S(M2,R)→ S(M1 ×M2,R) (4.4)f1 ⊗ f2 7→ {f1 f2 : (x, y) 7→ f1(x)f2(y)}extends to an isomorphism of TVS:S(M1,R) ⊗̂ S(M2,R) ∼−→ S(M1 ×M2,R) (4.5)4.2.3 Complex-Valued Schwartz FunctionsLet M be an affine Nash manifold. We fix an R-basis {1, i} of C. Eachfunction f ∈ C∞(M) is written as f = (f1, f2) where f1 is the real part andf2 is the imaginary part. We have the canonical isomorphismC∞(M,R)⊗R C→ C∞(M,C) (4.6)f ⊗ z 7→ {x 7→ f(x)z}85which is actually independent of the choice of the R-basis. This isomorphismmeans a complex valued function is smooth if and only if its real and imagi-nary parts are smooth. The ring DiffNash(M) of Nash differential operatorsacts on C∞(M,R), hence also on the C∞(M,C).Definition 4.17. A function f ∈ C∞(M,C) is a Schwartz (C-valued)function on M , if for all D ∈ DiffNash(M), the Df is bounded on M . Wedenote the space of Schwartz C-valued functions byS(M).We endow this space with the TVS structure induced by the semi-norms| · |D:|f |D := supx∈M|(Df)(x)|Let DiffNash(M) ⊗R C be the (complexified) algebra of complex Nashdifferential operators. It acts on the C∞(M,C) = C∞(M,R) ⊗R C in thenatural way. We haveLemma 4.18. A function f ∈ C∞(M,C) is in S(M) if and only if Df isbounded on M for all D ∈ DiffNash(M)⊗R C.Lemma 4.19. For the space S(M) of C-valued Schwartz functions, thenatural mapS(M,R)⊗R C→ S(M) (4.7)f ⊗ z 7→ {x 7→ f(x)z}is an isomorphism between TVS. In particular, the S(M) is a (complex)nuclear Fre´chet space.Remark 4.20. By this crucial isomorphism, we can freely generalize theproperties of S(M,R) to S(M). Actually the maps (4.2), (4.3) and (4.4)all extends to the spaces of C-valued Schwartz functions, since the functor−⊗R C is exact on NF-spaces.We have C-valued analogues of Lemma 4.9, 4.10, 4.12, 4.15 and 4.16.We summarize them as the following Lemma:Lemma 4.21. Let M be an affine Nash manifold, and for each restrictedopen subset U ⊂ M , let S(U) be the space of C-valued Schwartz functionson the affine Nash manifold U .86(C− 1) The space S(U) is a nuclear Fre´chet space over C for each restrictedopen subset U .(C− 2) For each pair U1, U2 of restricted open subsets of M such that U1 ⊂U2, the extension map (by zero)exU2U1 : S(U1)→ S(U2)is an injective homomorphism of TVS, with image equal to theclosed subspace{f ∈ S(U2) : Df ≡ 0 on U2 − U1,∀D ∈ DiffNash(U2)}.(C− 3) For each finite restricted open cover {Ui}ki=1 of an affine Nash mani-fold M , one can choose a smooth function αi ∈ C∞(M,C) such thatsuppαi ⊂ Ui,∑ki=1 αi ≡ 1 and αif ∈ S(Ui) for all f ∈ S(M).(C− 4) The correspondence U 7→ S(U) with the above extension maps forma cosheaf (of NF-spaces) on the restricted topology of M .(C− 5) For each closed affine Nash submanifold Z ⊂ M , the restrictionmapS(M)→ S(Z), f 7→ f |Zis a surjective homomorphism of TVS.(C− 6) Let M1,M2 be two affine Nash manifolds, then the external tensorproduct induces an isomorphism of TVS:S(M1) ⊗̂ S(M2) ∼−→ S(M1 ×M2).4.3 Vector Valued Schwartz Functions4.3.1 Vector Valued Schwartz Functions on Affine NashManifoldsIn this subsection, letM = an affine Nash manifoldE = a nuclear Fre´chet TVS over CWe introduce the notion of E-valued Schwartz functions on M .87Definition of E-Valued Schwartz FunctionsThe E-valued Schwartz functions are defined in the similar way as R-valued or C-valued Schwartz functions.Definition 4.22. A smooth function f ∈ C∞(M,E) is called a SchwartzE-valued function on M , if for all D ∈ DiffNash(M), the derivative Dfis bounded on M . More precisely, let {| · |ρ : ρ ∈ P} be the family of semi-norms defining the Fre´chet structure of E, then a function f ∈ C∞(M,E) isa Schwartz E-valued function if for all semi-norm |·|ρ, and D ∈ DiffNash(M),one hassupx∈M|(Df)(x)|ρ <∞.We denote byS(M,E)the space of Schwartz E-valued functions on M , and endow it with thetopology defined by seminorms {qD,ρ : D ∈ DiffNash(M), ρ ∈ P} whereqD,ρ(f) := supx∈M|(Df)(x)|ρ.Lemma 4.23. The S(M,E) is a nuclear Fre´chet space, under the topologydefined by seminorms {qD,ρ : D ∈ DiffNash(M), ρ ∈ P}.Lemma 4.24 ([20] p268 Proposition 1.2.6). The natrual mapS(M)⊗ E → S(M,E) (4.8)f ⊗ v 7→ {x 7→ f(x)v}is a continuous linear map which extends to an isomorphism of TVSS(M) ⊗̂ E ∼−→ S(M,E) (4.9)Open Extensions of Schwartz E-Valued FunctionsLet M be an affine Nash manifold, and U be a restricted open subset ofM . Then U is also an affine Nash manifold, and we have the spaces S(U)and S(U,E) of Schwartz functions on U .Given a f ∈ S(U,E), let exXU f be the naive extension of f to M by zero:exMU f(x) :={f(x), if x ∈ U0, if x ∈M − U88Lemma 4.25. The exXU f is in S(X,E). And the mapexXU : S(U,E)→ S(X,E) (4.10)f 7→ exXU fis an injective continuous linear map between TVS. Its image is the subspace{f ∈ S(M,E)|(Df)|M−U = 0, ∀D ∈ DiffNash(M)}which is a closed subspace. Hence exMU is a homomorphism of TVS.By Lemma 4.24, the map exMU maps the subspace S(U)⊗ E of S(U,E)continuously to the subspace S(M)⊗E, hence extends to a continuous linearmap on the completions S(U,E) → S(M,E). Obviously this extension isexactly the exMU .Partition of UnityLemma 4.26. Let M be an affine Nash manifold, and {Ui}ki=1 be a finiterestricted open cover of M . There exists a partition of unity, i.e. a smoothfunction αi ∈ C∞(M,R), suppαi ⊂ Ui and∑ki=1 αi ≡ 1 on M . Moreover,for each f ∈ S(M,E), we have αif ∈ S(Ui, E).We can use the same partition of unity as in Lemma 4.10.Cosheaf Property of S(−, E)Lemma 4.27. The correspondence S(−, E) : U 7→ S(U,E) is a cosheaf ofNF-spaces on the restricted topology of M .Given a finite restricted open cover U = ∪ni=1Ui of a restricted opensubset U , the sequencen−1∏i=1n∏j=i+1S(Ui ∩ Uj)→n∏i=1S(Ui)→ S(U)→ 0is exact since S(−) is a cosheaf on the restricted topology. Since E is nuclearFre´chet, the functor − ⊗̂ E is exact. We obtain the exact sequencen−1∏i=1n∏j=i+1S(Ui ∩ Uj , E)→n∏i=1S(Ui, E)→ S(U,E)→ 0by Lemma 4.24.89Closed Restrictions of Schwartz E-Valued FnctionsLemma 4.28. Let M be an affine Nash manifold, Z ⊂M be a closed Nashsubmanifold. The restriction mapS(M,E)→ S(Z,E) (4.11)f 7→ f |Zis a surjective homomorphism of TVS.The restriction f |Z is obviously Schwartz. The above map is obtainedby tensoring the map S(M,C) → S(Z,C) in Lemma 4.21, by the E. SinceE is nuclear, the − ⊗̂ E is exact, and the above map is still surjective.Externel Tensor ProductsLemma 4.29. Let M1,M2 be two affine Nash manifolds, E1, E2 be twonuclear Fre´chet spaces. The natural mapS(M1, E1)⊗ S(M2, E2)→ S(M1 ×M2, E1 ⊗̂ E2) (4.12)f1 ⊗ f2 7→ {f1 f2 : (x1, x2) 7→ f1(x1)⊗ f2(x2)}is continuous linear and extends to an isomorphismS(M1, E1) ⊗̂ S(M2, E2) ∼−→ S(M1 ×M2, E1 ⊗̂ E2). (4.13)By Lemma 4.24, we have the natural mapS(M1)⊗ S(M2)⊗ E1 ⊗ E2 → S(M1 ×M2, E1 ⊗̂ E2)which extends to an isomorphism on the completion.4.3.2 Summary of Properties of Schwartz FunctionsWe summarize the crucial properties about Schwartz E-valued functionsin Lemma 4.23, 4.24, 4.25, 4.26, 4.27, 4.28 and 4.29, as the following propo-sition. The Lemma 4.21 is thus the special case of the following propositionwhen E = C.Proposition 4.30. Let M be an affine Nash manifold, E be a nuclearFre´chet space, U be an arbitrary restricted open subset of M . Let S(M)(resp. S(U)) be the space of C-valued Schwartz functions on M (resp. U),and S(M,E) (resp. S(U,E)) be the space of E-valued Schwartz functionson M (resp. U).90(E − 1) For each restricted open U , the S(U,E) is a nuclear Fre´chet TVS,and the natural map S(U) ⊗ E → S(U,E) extends to an isomor-phism of TVS:S(U) ⊗̂ E ∼−→ S(U,E).(E − 2) For a restricted open subset U ⊂M , the extension map (by zero)S(U,E) ↪→ S(X,E) (4.14)is an injective homomorphism of TVS.(E − 3) Let {Ui}ki=1 be a restricted open cover of M , then there exists apartition of unity subordinate to the cover, i.e. there exist smoothfunctions αi on M such that suppαi ⊂ Ui,∑ki=1 αi ≡ 1 and αif ∈S(Ui, E) for all f ∈ S(M,E).(E − 4) The correspondence U 7→ S(U,E) is a cosheaf of NF-spaces on therestricted topology of M .(E − 5) Let Z ⊂M be a closed Nash submanifold of M and S(Z,E) be thespace of E-valued Schwartz functions on Z. The restriction mapS(X,E) S(Z,E) (4.15)is a surjective homomorphism of TVS.(E − 6) Let M1,M2 be two smooth real algebraic manifolds, let E1, E2 betwo nuclear Fre´chet spaces, and S(Xi, Ei), i = 1, 2 be the spaces ofEi-valued Schwartz functions on Mi. The natural mapS(M1, E1) ⊗̂ S(M2, E2) ∼−→ S(M1 ×M2, E1 ⊗̂ E2). (4.16)is an isomorphism of TVS.4.4 Schwartz DistributionsIn this section, letM = an affine Nash manifoldE,F = two nuclear Fre´chet spacesS(−, E) = the cosheaf of Schwartz E-valued functionson the restricted topology of M91Definition 4.31. A continuous linear map from S(M,E) to F is called aSchwartz F -valued E-distribution on M , and we denote byL(S(M,E), F )the space of Schwartz F -valued E-distributions on M . It is exactlythe space of continuous linear maps from S(M,E) to F .If F = C, we simply call an element Φ ∈ L(S(M,E),C) a SchwartzE-distribution on M , and the space of Schwartz E-distributions on M isalso denoted byS(M,E)′ = L(S(M,E),C)i.e. it is exactly the strong dual of S(M,E) with strong topology.If further E = F = C, we simply call an element Φ ∈ L(S(M),C) aSchwartz distribution on M , and the space of Schwartz distributions onM is exactly strong dual space S(M)′.By abuse of terms, we simply call them distributions, when there is noambiguity on the spaces E and F .Since S(M,E) and F are both nuclear Fre´chet, by (50.18) on page 525of [36], we haveLemma 4.32. The natural map S(M,E)′⊗F → L(S(M,E), F ) extends toan isomorphism of TVS:S(M,E)′ ⊗̂ F ∼−→ L(S(M,E), F ) (4.17)In particular, the space L(S(M,E), F ) is a tensor product of an NF-spaceF with a DNF-space S(M,E)′, hence is a nuclear space.Remark 4.33. Note that in general L(S(M,E), F ) is not a Fre´chet space.For example, when F = C, the distribution space is S(M,E)′–a dual of aFre´chet space, which is almost never Fre´chet unless it is finite dimensional(e.g. M is a point, and E is finite dimensional, then the S(M,E) = E andS(M,E)′ = E′.)4.4.1 Sheaf of Schwartz DistributionsWe keep the M,E,F as the beginning of this section.Definition 4.34. Let U1, U2 be two restricted open subsets of M such thatU1 ⊂ U2. We have the open extension map as in Lemma 4.25exU2U1 : S(U1, E) ↪→ S(U2, E).92The F -transpose of this extension map gives the following continuous linearmapresU2U1 : L(S(U2, E), F )→ L(S(U1, E), F ) (4.18)D 7→ D ◦ exU2U1called the restriction map of Schwartz distributions from U2 to U1.The restriction maps are homomorphisms of TVS, and they are all sur-jective homomorphisms, i.e. all local Schwartz distributions are restrictionsof global Schwartz distributions.The correspondence U 7→ L(S(U,E), F ) is a presheaf (of nuclear TVS)on the restricted topology of M . Actually we haveLemma 4.35. The correspondence U 7→ L(S(U,E), F ) is sheaf on the re-stricted topology on M .Proof. By Lemma 4.32, we just need to show the case F = C. The “sheaf-sequence” is exact because it is the dual sequence of the “cosheaf-sequence”,which is exact.4.4.2 Extension of Schwartz Distributions from ClosedNash SubmanifoldsLet M,E,F be as the beginning of this section. Let Z ⊂M be a closedNash submanifold. By Proposition 4.30, we have the mapS(M,E) S(Z,E), f 7→ f |Zwhich is a surjective homomorphism of TVS. Its F -transpose gives a homo-morphism on the space of F -valued Schwartz E-distributions:Lemma 4.36. The F -transpose map of the homomorphism S(M,E) S(Z,E) (surjective) is an injective homomorphism of TVS:L(S(Z,E), F ) ↪→ L(S(M,E), F ) (4.19)called the extension of Schwartz distributions from Z to M .934.4.3 Distributions Supported in Closed SubsetsDefinition 4.37. A subset Z of a affine Nash manifold M is called a re-stricted closed subset, if its complement M − Z is a restricted opensubset.Let U ⊂M be a restricted open subset, Z = M −U be its complement,(a restricted closed subset of M) and D ∈ L(S(M,E), F ) be a SchwartzF -valued E-distribution. We say D is supported in Z, or D vanishes onU , if resMU (D) = 0, i.e. the restriction of D to U is zero. The space of dis-tributions supported in Z (vanishing on U) is exactly the kernel Ker(resMU ),which is a closed subspace of L(S(M,E), F ).Lemma 4.38. Let U1, U2 be two restricted open subsets of M such thatU1 ⊂ U2, let Z be a restricted closed subset of U1 (the U1 − Z and U2 − Zare restricted open subsets of M). Then the restriction mapresU2U1 : L(S(U2, E), F )→ L(S(U1, E), F )sends Ker(resU2U2−Z) isomorphically to Ker(resU1U1−Z).The open embeddingsU1 U2U1 − Z U2 − Zinduce open extension mapsS(U1, E) S(U2, E)S(U1 − Z,E) S(U2 − Z,E)The F -transpose of this diagram thus gives the following diagram of restric-tion maps of distributions:94Ker(resU1U1−Z) Ker(resU2U2−Z)L(S(U1, E), F ) L(S(U2, E), F )L(S(U1 − Z,E), F ) L(S(U2 − Z,E), F )resU2U1resU2−ZU1−ZresU1U1−Z resU2U2−ZFigure 4.1: Independent of NeighbourhoodsThis Lemma says: the top horizontal arrow is an isomorphism of TVS.Proof. (Injectivity): Let D ∈ Ker(resU2U2−Z) be a Schwartz distribution onU2 vanishing on U2 − Z. If resU2U1(D) = 0, then D satisfiesresU2U1(D) = 0, resU2U2−Z(D) = 0.Note that U1∪(U2−Z) = U2, by the sheaf property of L(S(−, E), F ), we seeD = 0 as a distribution on U2. Hence resU2U1maps Ker(resU2U2−Z) injectivelyto Ker(resU1U1−Z).(Surjectivity): Let D ∈ Ker(resU1U1−Z) ⊂ L(S(U1, E), F ) be a Schwartzdistribution on U1 vanishing on U1−Z. Consider the zero distribution 0U2−Zon the restricted open subset U2 − Z. Obviously we haveresU1U1−Z(D) = 0 = resU2−ZU1−Z(0U2−Z)i.e. the D and 0U2−Z agree on the restricted open subset U1−Z = U1∩(U2−Z). By the sheaf property, they glue up to a distribution D˜ on U1∪(U2−Z) =U2. Obviously resU2U1(D˜) = D, and resU2U2−Z(D˜) = 0, hence we find a pre-image D˜ of D in Ker(resU2U2−Z).As a corollary to the above Lemma, we haveLemma 4.39. Let U1, U2 be two restricted open subsets of M , and let Z ⊂M be a subset of U1 and U2 which is restricted open in both of them. Thenthe two kernels Ker(resU1U1−Z) and Ker(resU2U2−Z) are isomorphic.95Proof. We just need to apply the above Lemma to U1, U2 and U1 ∩ U2, andthe following three kernels are canonically isomorphic:Ker(resU1U1−Z) ' Ker(resU1∩U2U1∩U2−Z) ' Ker(resU2U2−Z).4.5 Application to Nonsingular Affine RealAlgebraic VarietiesWe will apply the above theory of Schwartz functions and distributions,to nonsingular affine real algebraic varieties, or more specifically the G =G(R).In [20], Fokko du Cloux has defined the Schwartz functions on ”semi-algebraic varieties”. For the particular case of affine algebraic varieties, thedefinition of Schwartz functions in [1] coincide with the definition in [20]([1] p6 Remark 1.6.3). Hence we can identify the two notions of R-valuedSchwartz functions in [1] and [20].Remark 4.40. After the Chapter 4 is written, we found out that B. Elazarand A. Shaviv have generalized the work of Aizenbud-Gourevitch to (non-affine) real algebraic varieties (arXiv 1701.07334[math.AG]).4.5.1 Pseudo-Sheaf and Pseudo-CosheafIn this subsection, we introduce two terms: pseudo-sheaves and pseudo-cosheaves, on topological spaces. In a word, they are just sheaves/cosheaveswith the open covers in the sheaf/cosheaf axioms replaced by finite opencovers.Let(X, T ) = a topological space,i.e. X is a set and T is a topology on X.The topology T is a partial orderd set which could be regarded as acategory, with open subsets as objects, open inclusions as morphisms, theempty set ∅ as initial object and X as the final object.Let C be an abelian category. One can define the notion of presheavesand precosheaves on (X, T ) in the ordinary sense:Definition 4.41. A presheaf on (X, T ) with value in C is a contravariantfunctor from the category T to C. When there is no ambiguity on the96topology and category C, we simply call them presheaves. More precisely,let F be a presheaf, then• Each open subset U ∈ T is associated with an object F(U) ∈ C, andin particular F(∅) = {0}.• For each pair of open subset U1 ⊂ U2, one has a morphismresU2U1 : F(U2)→ F(U1)in C called the restriction from U2 to U1. And for three open subsetsU1 ⊂ U2 ⊂ U3, the restriction morphisms satisfyresU2U1 ◦ resU3U2 = resU3U1 .A precosheaf on (X, T ) with value in C is a covariant functor fromthe category T to C. When there is no ambiguity on the topology and C,we simply call them precosheaves. More precisely, let E be a precosheaf,then• Each open subset U ∈ T is associated with an object E(U) ∈ C, andin particular E(∅) = {0}.• For each pair of open subset U1 ⊂ U2, one has a morphismexU2U1 : E(U1)→ E(U2)in C called the extension from U1 to U2. And for three open subsetsU1 ⊂ U2 ⊂ U3, the restriction morphisms satisfyexU2U3 ◦ exU1U2 = exU1U3 .Definition 4.42. Let F be a presheaf on (X, T ) with values in C. We sayF is a pseudo-sheaf, if it satisfies the following two axioms:• Let U be an open subset and {Ui}ni=1 be a finite open cover of U :U = ∪ni=1Ui. Let f ∈ F(U) be a section on U . If resUUif = 0 for alli = 1, . . . , n, then f = 0.• Let U and {Ui}ni=1 be as above (i.e. finite cover). For each i, letfi ∈ F(Ui) be a section on Ui. If resUiUi∩Ujfi = resUjUi∩Ujfj for all1 ≤ i, j ≤ n, then there is a f ∈ F(U) such that fi = resUUif .97Equivalently, a presheaf F is a pseudo-sheaf, if for each open U and finiteopen cover {Ui}ni=1 of U , the following sequence is exact in C:0→ F(U) Φ1−→n∏i=1F(Ui) Φ2−→n−1∏i=1n∏j=i+1F(Ui ∩ Uj). (4.20)Here the map Φ1 isΦ1 : F(U)→n∏i=1F(Ui) (4.21)f 7→ (resUUif)ni=1and the map Φ2 isΦ2 :n∏i=1F(Ui)→n−1∏i=1n∏j=i+1F(Ui ∩ Uj) (4.22)(fi)ni=1 7→ ((resUiUi∩Ujfi − resUjUi∩Ujfj)nj=i+1)n−1i=1Definition 4.43. Let E be a precosheaf on (X, T ) with values in C. We sayE is a pseudo-cosheaf, if it satisfies the following two axioms:• Let U be an open subset and {Ui}ni=1 be a finite open cover of U :U = ∪ni=1Ui. Let f ∈ E(U) be a section on U . There exists a fi ∈ E(Ui)for each i, such thatf =n∑i=1exUUifi.• Let U and {Ui}ni=1 be as above (i.e. finite cover). For each i, letfi ∈ E(Ui) be a section on Ui. If∑ni=1 exUUifi = 0, then there exists afij ∈ E(Ui ∩ Uj) for all 1 ≤ i < j ≤ n, such thatfi = −∑1≤k<iexUiUk∩Uifki +∑i<k≤nexUiUi∩Ukfikfor each 1 ≤ i ≤ n.Equivalently, a precosheaf E is a pseudo-sheaf, if for each open U andfinite open cover {Ui}ni=1 of U , the following sequence is exact in C:n−1⊕i=1n⊕j=i+1E(Ui ∩ Uj) Ψ2−−→n⊕k=1E(Uk) Ψ1−−→ E(U)→ 0. (4.23)98Here the map Ψ1 isΨ1 :n⊕k=1E(Uk)→ E(U) (4.24)(fk)nk=1 7→n∑k=1exUUkfkand the map Ψ2 isΨ2 :n−1⊕i=1n⊕j=i+1E(Ui ∩ Uj)→n⊕k=1E(Uk) (4.25)((fij)nj=i+1)n−1i=1 7→ (−∑1≤i<kexUkUi∩Ukfik +∑k<j≤nexUkUk∩Ujfkj)nk=1Remark 4.44. • Note that even for a presheaf F , the sequence (4.20)is a complex, and for a precosheaf E , the sequence (4.23) is a complex.• We call them “pseudo”, since they only satisfies the axioms of sheavesand cosheaves on finite covers.4.5.2 Schwartz Functions and Distributions on NonsingularAffine Real Algebraic VarietiesLet X be a nonsingular affine real algebraic variety. As in 4.1.2, we haveseen that it has a natural structure of affine Nash manifold, and all Zariskiopen subsets of X are restricted open, i.e. the Zariski topology is includedin the restricted topology:T ZarX ⊂ T ResX .Note that the restricted topology T ResX is not a topology since infinite unionsof restricted open subsets need not be restricted open. However, the Zariskitopology T ZarX is indeed a topology.For two nuclear Fre´chet spaces E,F , we haveS(−, E) = the cosheaf of Schwartz E-valued functions on XL(S(−, E), F ) = the sheaf of Schwartz F -valued E-distributions on XThese cosheaf and sheaf are on the restricted topology T ResX of X, and wecan restrict them to the Zariski topology T ZarX . With the terms “pseudo-cosheaf” and “pseudo-sheaf” defined in 4.5.1, we have99Lemma 4.45. • The cosheaf S(−, E) on T ResX restricted to the Zariskitopology T ZarX , is a pseudo-cosheaf on T ZarX .• The sheaf L(S(−, E), F ) on T ResX restricted to the Zariski topologyT ZarX , is a pseudo-sheaf on T ZarX .Remark 4.46. In this thesis, we will only apply the Schwartz analysis tononsingular affine real algebraic varieties, i.e. the G = G(R) and its Zariskiopen/closed subvarieties. We will restrict the cosheaf of Schwartz functionsand sheaf of Schwartz distributions to the Zariski topologies, and regardthem as pseudo-cosheaves/pseudo-sheaves.4.5.3 Supports and Maximal Vanishing Subsets ofDistributionsAs in 4.5.2, let X be a nonsingular affine real algebraic variety, and wedefine the cosheaves of Schwartz functions and sheaves of Schwartz distribu-tions on it, by regarding it as an affine Nash manifold. As in Lemma 4.45,we restrict the cosheaves of Schwartz functions and sheaves of Schwartz dis-tributions to the Zariski topology of X and obtain the pseudo-cosheaves ofSchwartz functions and pseudo-sheaves of Schwartz distributions.An advantage of restricting the sheaf of Schwartz distributions to theZariski topology is—such distributions do have supports on the Zariskitopology.Remark 4.47. In classical distribution analysis, the support of a distribu-tion D ∈ C∞c (M)′ on a smooth manifold M , is defined to be the complementof the maximal vanishing open subset of D (see [36] p255 Corollary 2 andDefinition 24.2). More precisely, let U ⊂ M be the maximal (Euclidean)open subset of M such that 〈D, f〉 = 0 for all f ∈ C∞c (U). Then the sup-port of D denoted by suppD is defined to be the complement of U . Herethe maximal vanishing open subset of D is obtained by taking union of allopen subsets on which D vanishes. And by Corollary 2 on p255 of [36], thedistribution D vanishes on this union.Note that on restricted topology, an infinite union of restricted opensubsets may not be a restricted open subset. Hence one cannot define “thesupport” of a Schwartz distribution as in classical distribution analysis. Andwe only have the notion “support in” as in Definition 4.37.However, if we restrict the sheaf of Schwartz distributions to the Zariskitopology, we do have the notion of support of a Schwartz distribution, since100the Zariski topology is indeed a topology on which one can take infiniteunions.Definition 4.48. Let X be a nonsingular affine real algebraic variety, andE,F be two nuclear Fre´chet spaces. We regard S(−, E) as a pseudo-cosheafon X and L(S(−, E), F ) as a pseudo-sheaf on X.Let D ∈ L(S(X,E), F ) be a Schwartz F -valued E-distribution on X.LetvanD = the union of all Zariski open subsets U ⊂ Xsuch that resXU (D) = 0suppD = (vanD)c= the complement of vanDWe call vanD the maximal vanishing subset of D and suppD the sup-port of D.Lemma 4.49. For D ∈ L(S(X,E), F ), the vanD is a Zariski open subsetof X, and suppD is a Zariski closed subset of X. The vanD is the largestZariski open subset of X on which D vanishes, i.e. the name “maximalvanishing subset” is reasonable.Proof. The first part is trivial. We just need to show D indeed vanishes onvanD.By definition, vanD is the union of all Zariski open U such that D|U = 0.Hence vanD is covered by {U ∈ T ZarX : D|U = 0}. The Zariski topologyis quasi-compact, hence we can choose a finite subcover of vanD from thefamily {U ∈ T ZarX : D|U = 0}, say {Ui}mi=1. Hence D|Ui = 0 for i = 1, . . . ,mand vanD = ∪mi=1Ui. By the partition of unity property, we can find αismooth functions on Ui, such that suppαi ⊂ Ui and∑mi=1 αi ≡ 1. Then foran arbitrary f ∈ S(vanD,E), we have〈D, f〉 = 〈D,m∑i=1αif〉 =m∑i=1〈resXUi(D), αif〉 = 0.Hence D|vanD = 0, i.e. D vanishes on vanD.4.5.4 Geometry on G = G(R)We list some well-known facts about G = G(R), and its subspaces (sub-varieties/submanifolds). Note that G = G(R) is a101• Real affine algebraic variety;• Smooth manifold (Lie group);• Affine Nash manifold.G as a Real Algebraic VarietyThe algebraic group G is defined over R, hence is R-closed. Since G isaffine, this means we can embed G into a general linear group GLn ⊂ An2 asan closed subvariety, and the defining ideal of G is generated by polynomialswith real coefficients.Lemma 4.50. For the group G = G(R), we have• G is an affine real algebraic variety.• For any R-closed subgroup H of G, the H = H(R) is an affine realalgebraic variety, and Zariski closed subset in G.• Let P,Q be two R-closed subgroups of G, and P = P(R), Q = Q(R).Then every (P,Q)-double coset on G is locally closed under Zariskitopology, hence is an affine real algebraic variety.• Let P,Q be as above, and assume there are finitely many (P,Q)-doublecosets on G parameterized by a finite set I. For each i ∈ I, the Zariskiopen subsets G≥i, G>i are affine real algebraic varieties.G as a Smooth ManifoldWe regard G = G(R) as a Lie group. Let G ⊂ GLn be an embeddingof the variety G into general linear group (define over R). This induces theembedding on real points G = G(R) ⊂ GL(n,R) ⊂ Rn2 , which gives thecanonical smooth structure on G.Lemma 4.51. For the group G = G(R) with the canonical structure ofsmooth manifold, we have• G = G(R) is a regular submanifold of GL(n,R) (and Rn2).• For R-closed subgroup H of G, the H = H(R) is a regular closedsubmanifold of G (also of GL(n,R)).102• For P,Q be two R-closed subgroup of G, and P = P(R), Q = Q(R).The (P,Q)-double cosets on G are locally closed regular submanifoldsof G.• Let P,Q be as above, and assume there are finitely many (P,Q)-doublecosets on G parameterized by I. Then for each i ∈ I, the Zariski opensubsets G≥i, G>i are open under the manifold (Euclidean) topology andthey are open regular submanifolds of G.G as an Affine Nash ManifoldEvery nonsingular affine real algebraic variety has a canonical structureof affine Nash manifold. Combining the above two subsections, we can studyG = G(R) as an affine Nash manifold.Lemma 4.52. For G = G(R), we have• The G ⊂ Rn2 is a closed Nash submanifold, hence an affine Nashmanifold.• For H ⊂ G a R-closed subgroup, the H = H(R) is a closed affine Nashsubmanifold of G.• For P,Q two R-closed subgroups of G, and P = P(R), Q = Q(R),each (P,Q)-double coset is a affine Nash manifold.• For P,Q as above, assume there are finitely many (P,Q)-double cosetson G, parameterized by the finite set I. Then for each i ∈ I, theG≥i, G>i are affine Nash manifolds.4.5.5 Right Regular Actions on Schwartz Function SpacesIn this subsection, we let G = G(R) be the real point group of a con-nected reductive linear algebraic group G defined over R. It is a nonsingularaffine real algebraic variety and an affine Nash manifold. LetE = a nuclear Fre´chet spaceH = a R-closed subgroup of GY = a nonsingular R-subvariety of Gstable under the right H-translation103Then H = H(R) is a Lie subgroup of G, Y = Y(R) is a real subvariety ofG stable under the right H-translation. And the H acts algebraically on Yby the right translation.The Y is nonsingular affine real algebraic variety, and it has the spaceS(Y,E) of Schwartz E-valued functions on it.Definition 4.53 (Right regular actions on Schwartz spaces). Theright regular H-action on the space S(Y,E) of Schwartz E-valued func-tions on Y , is defined to be[Rhf ](y) := f(yh), ∀h ∈ H, y ∈ Y, f ∈ S(Y,E).Lemma 4.54. For the right regular H-action on S(Y,E), we have:1. The S(Y,E) is a smooth H-representation. It is isomorphic to thetensor product representation S(Y ) ⊗̂ E, where S(Y ) is the space ofC-valued Schwartz functions on Y with the right regular H-action, Eis considered as a trivial H-representation.2. Let U1, U2 be two right H-stable Zariski open subsets of G, and U1 ⊂U2. The inclusion mapS(U1, E) ↪→ S(U2, E)is H-equivariant, i.e. an H-intertwining operator.3. Let U be a right H-stable Zariski open subset of G, O be a right H-stable nonsingular Zariski closed subset of U . The restriction mapS(U,E)→ S(O,E)is H-equivariant, i.e. an H-intertwining operator.Example 4.55. For H = P∅, Y = G or G≥w, G>w,∀w ∈ [WΘ\W ], thespaces S(G,E),S(G≥w, E) and S(G>w, E) have the above right regular P∅-actions, and they are smooth P∅-representations. The following inclusionmaps are all P∅-equivariant (intertwining operators):S(G>w, E) ↪→ S(G≥w, E) ↪→ S(G,E).Example 4.56. For H = P , Y = G or P , the S(G,E) and S(P,E) aresmooth P -representations. The surjective restriction mapS(G,E)→ S(P,E)104is P -equivariant.For H = P∅, Y = G≥w or PwP∅, the S(G≥w, E) and S(PwP∅, E) aresmooth P∅-representations, and the surjective restriction mapS(G≥w, E)→ S(PwP∅, E)is P∅-equivariant.4.6 Schwartz InductionsIn this section, we recall the notion of Schwartz inductions, and showsome properties on such spaces and their strong dual.The Schwartz induction is first introduced in section 2 of [20]. We haveseen that notion of Schwartz functions defined in [20] is the same as theSchwartz functions defined in [1], in case the manifolds under considera-tion are nonsingular affine real algebraic varieties. Hence we could combinethe works in [20] and [1], to prove parallel a set of properties on Schwartzinductions, as summarized in Proposition 4.30.4.6.1 Schwartz Induction SIndGPσLetG = a linear algebraic group defined over RP = a R-(closed) subgroup of GG = G(R) = the Lie group of real points of GP = P(R) = the Lie group of real points of Pdp = a fixed right invariant measure on P(σ, V ) = a Harish-Chandra representation of P(In particular V is Fre´chet and of moderate growth)S(G,V ) = the space of V -valued Schwartz functions on GThe σ-Mean Value and the Schwartz InductionFor a function f ∈ S(G,V ), we can define its σ-mean value byfσ(g) :=∫Pσ(p)−1f(pg)dp, ∀g ∈ G. (4.26)This integration converges, since the f is Schwartz and σ is of moderategrowth.105The fσ is a smooth V -valued function on G, satisfying the following“σ-rule”:fσ(pg) = σ(p)fσ(g), ∀p ∈ P,∀g ∈ G.In other words, the fσ is in the smooth induction space C∞IndGPσ.The map S(G,V ) → C∞IndGPσ, f 7→ fσ is continuous linear and G-equivariant when S(G,V ) and C∞IndGPσ are both endowed with right reg-ular G-actions.Definition 4.57 ([20] p273 Definition 2.1.2). We denote the image of themap f 7→ fσ in C∞IndGPσ bySIndGP (σ, V )or simply SIndGPσ if there is no ambiguity. We endow the space SIndGP (σ, V )with the quotient topology from S(G,V ). The SIndGP (σ, V ) is called theSchwartz induction space of (σ, V ) (from P to G).Since a quotient of a NF-space is still a NF-space, we have:Lemma 4.58. The SIndGP (σ, V ) is a nuclear Fre´chet space, and it is asmooth representation of G.Schwartz Inductions vs. Smooth InductionsBy definition, the Schwartz induction space SIndGPσ is a subspace of thesmooth induction space C∞IndGPσ. By a partition of unity argument, wesee the spaceC∞c IndGPσ = smooth induction with compact support modulo Pis contained in SIndGPσ, hence one has the following inclusions:C∞c IndGPσ ⊂ SIndGPσ ⊂ C∞IndGPσ.Actually, if the quotient manifold P\G is compact, the above inclusionsare equalities:Lemma 4.59 ([20] p273 Remark 2.1.4). If the quotient manifold P\G iscompact, then the three spaces of inductions coincide:C∞c IndGPσ = SIndGPσ = C∞IndGPσ.In particular, the SIndGP (σ, V ) is a Harish-Chandra representation.We will mainly apply this Lemma to the case of parabolic inductions.If P is a parabolic R-subgroup of G, then the P\G is complete, and themanifold of real points P\G is compact by [11] p146 Proposition 14.2.106Schwartz Inductions are NOT Schwartz on GThe functions in the Schwartz inductions are not necessarily rapidlydecreasing on G. We give some examples.Example 4.60. Let G = (R,+) be the additive group of real numbers, andlet P = G. The irreducible representations of P = R are characters of theformχλ : R→ C×r 7→ eλrfor some λ ∈ C. It is of moderate growth if and only if χλ is unitary orequivalently λ ∈ iR.Suppose λ ∈ iR, we consider the Schwartz induction SIndGPχλ. A func-tion φ in it is a smooth function satisfyingφ(r · x) = φ(x+ r) = χλ(r)φ(x) = eλrφ(x), ∀r ∈ P = R, x ∈ G = R.In particular, for arbitrary r ∈ R, we have φ(r) = eλrφ(0). We seelimr→+∞φ(r) = φ(0) limr→+∞ eλrlimr→−∞φ(r) = φ(0) limr→−∞ eλrIf the φ is Schwartz, then both limits should be zero. However if λ 6= 0, thelimits on the right-hand-side do not exist (periodic function with image onunit circle), if λ = 0, the limits equal to φ(0) which may not be zero.Example 4.61. Let G = (R2,+) be the additive group of real plane (withcoordinates (x, y)), and let P = Ry be the y-axis. The P is a closed subgroupof G, with embedding y 7→ (0, y). For a λ ∈ iR, let χλ be a character of Pas the above example.Let φ ∈ SIndGPχλ be an arbitrary function in the Schwartz induction.Then it is a smooth function on G = R2 satisfyingφ(r · (x, y)) = φ(x, y + r) = eλrφ(x, y).In particular, φ(0, r) = eλrφ(0, 0), and we see the restriction of φ to Ry is nota Schwartz function. However, for every fixed y, the φ(x, y) as a function ofx, is still rapidly decreasing.107Example 4.62. In general, let G be the real point group of a connectedreductive linear algebraic group defined over R, P be the real point groupof a parabolic subgroup. Let σ be the one dimensional trivial character ofP . Then SIndGPσ is not contained in S(G,C). Suppose not, and assume aφ ∈ SIndGPσ is a Schwartz function on G, then its restriction to P is still aSchwartz function (by (E-5) of Proposition 4.30). However,φ(p) = σ(p)φ(e) = φ(e), ∀p ∈ P.Hence φ restricted to P is a constant function, which is not Schwartz on P .Remark 4.63. Although functions in SIndGPσ are not rapidly decreasingon the entire G, we will see later that they are rapidly decreasing along the“orthogonal direction” of P .4.6.2 Local Schwartz InductionsWe generalize the above construction and define the local Schwartz in-ductions. This notion is defined in 2.2.4 of [20]. We keep the setting as inlast subsection, and letY = a locally closed nonsingular subvariety of Gwhich is stable under left P -translation.Since Y is a nonsingular locally closed subvariety of G, one has the spaceS(Y, V ) of Schwartz V -valued functions on Y , and one can define the σ-meanvalue function as in (4.26), i.e. for a f ∈ S(Y, V ), and y ∈ Y letfσ(y) :=∫Pσ(p−1)f(py)dp (4.27)Then fσ is a smooth V -valued function on Y , satisfying the σ-rule:fσ(py) = σ(p)fσ(y), ∀p ∈ P, y ∈ Y.LetC∞(Y, V, σ) = {f ∈ C∞(G,V ) : f(py) = σ(p)f(y), ∀p ∈ P, y ∈ Y }be the space of smooth V -valued functions satisfying the σ-rule. The corre-spondence f 7→ fσ is a continuous linear map from S(Y, V ) to C∞(Y, V, σ).108Definition 4.64. The image of the map S(Y, V )→ C∞(Y, V, σ), f 7→ fσ isdenoted bySIndYP (σ, V )or simply SIndYP σ if there is no ambiguity, and we call it the local Schwartzinduction space of σ from P to Y . We endow this space with the quotienttopology from S(Y, V ), and one has a surjective homomorphism of TVS:S(Y, V ) SIndYP σ (4.28)f 7→ fσLemma 4.65. The SIndYP σ is a nuclear Fre´chet TVS.4.6.3 Open Extensions and Closed Restrictions of SchwartzInductionsSimilar to the Schwartz V -valued functions, we have the extension fromopen subsets and restriction to closed subsets.Open Extensions of Schwartz InductionsLet U1, U2 be two Zariski open subvarieties of G, which are stable underleft P -translation, and assume U1 ⊂ U2.The open embedding induces the injective homomorphism of TVS (ex-tension of the cosheaf):S(U1, V ) ↪→ S(U2, V ), f 7→ exU2U1f.Let φ ∈ SIndU1P σ, and let f ∈ S(U1, V ) be a Schwartz function such thatφ = fσ. Then the above embedding gives the function exU2U1f ∈ S(U2, V )(cosheaf extension). Then its σ-mean value function (exU2U1f)σ is in SIndU2P σ.We haveLemma 4.66. The (exU2U1f)σ is independent of the choice of f ∈ S(U1, V ),and it is exactly the extension of φ by zero. More precisely, letExU2U1φ(x) ={φ(x), if x ∈ U10, if x ∈ U2 − U1Then (exU2U1f)σ = ExU2U1φ for all f ∈ S(U1, V ) such that fσ = φ. Hence theExU2U1φ is in SIndU2P σ.109Proof. (1) We first verify that the (exU2U1f)σ is independent of the choice off . Let f0 ∈ S(U1, V ) satisfy (f0)σ = 0 in SIndU1P σ. Let exU2U1f0 ∈ S(U2, V )be its extension to U2, and let (exU2U1f0)σ ∈ SIndU2P σ be its σ-mean valuefunction.By definition, for all x ∈ U2(exU2U1f0)σ(x) =∫Pσ(p−1)exU2U1f0(px)dp.If x ∈ U1, since U1 is stable under left P -translation, the px ∈ U1 for allp ∈ P , and exU2U1f0(px) = f0(px). Hence(exU2U1f0)σ(x) =∫Pσ(p−1)f0(px)dp = (f0)σ(x) = 0.If x ∈ U2 − U1, then px ∈ U2 − U1 (the complement U2 − U1 is also leftP -stable), and exU2U1f0(px) = 0. Hence(exU2U1f0)σ(x) =∫Pσ(p−1)0dp = 0.In sum, if (f0)σ = 0, then (exU2U1f0)σ = 0. Hence the (exU2U1f)σ is inde-pendent of the choice of f .(2) Second we check (exU2U1f)σ = ExU2U1φ. Let f ∈ S(U1, V ) be a Schwartzfunction satisfying fσ = φ. Still for all x ∈ U2, one has(exU2U1f)σ(x) =∫Pσ(p−1)exU2U1f(px)dp.If x ∈ U1, then px ∈ U1,∀p ∈ P , and exU2U1f(px) = f(px) and(exU2U1f)σ(x) =∫Pσ(p−1)f(px)dp = fσ(x) = φ(x) = ExU2U1φ(x).If x ∈ U2 − U1, then px ∈ U2 − U1,∀p ∈ P , and exU2U1f(px) = 0 and(exU2U1f)σ(x) =∫Pσ(p−1)0dp = 0 = ExU2U1φ(x).110Lemma 4.67. The mapSIndU1P σ → SIndU2P σ (4.29)φ 7→ ExU2U1φis a continuous linear map and an injective homomorphism of TVS. Thishomomorphism makes the following diagram commute:S(U1, V ) S(U2, V )SIndU1P σ SIndU2P σProof. The map is obviously linear, and the above diagram commute. It iscontinuous because the topology on SIndU1P σ is the quotient topology, andthe composition mapS(U1, V )→ SIndU1P σ → SIndU2P σis continuous since the composition other way is continuous. It is injectivesince the map is extension by zero. It is a homomorphism of TVS since ithas closed image and all spaces are Fre´chet.Closed Restriction of Schwartz InductionsLet U be a Zariski open subvariety of G, and O be a Zariski closedsubvariety of U (locally closed in G). Assume U and O are all stable underleft P -translation. Let SIndUPσ,SIndOPσ be their Schwartz induction spaces.One has the surjective homomorphism of TVSS(U, V ) S(O, V ), f 7→ f |Ogiven by the restriction of Schwartz functions to O.Let φ ∈ SIndUPσ be an arbitrary element. Let f ∈ S(U, V ) be a Schwartzfunction such that fσ = φ, and let f |O ∈ S(O, V ) be its restriction to O.Its σ-mean value function (f |O)σ is in SIndOPσ. We haveLemma 4.68. The function (f |O)σ ∈ SIndOPσ is independent of the choiceof f ∈ S(U, V ). More precisely, letφ|O = the restriction of φ to O.Then φ|O = (f |O)σ for all f ∈ S(U, V ) such that fσ = φ. Hence the φ|O isin the Schwartz induction space SIndOPσ.111Proof. We just need to check the (f |O)σ = φ|O. Let f ∈ S(U, V ) be aSchwartz function such that fσ = φ in SIndUPσ. For x ∈ O, the(f |O)σ(x) =∫Pσ(p−1)[f |O](px)dp.Note that px ∈ O,∀p ∈ P , hence [f |O](px) = f(px). The above integrationis exactly fσ(x) for all x ∈ O. Hence as functions on O, the (f |O)σ is exactlythe restriction φ|O.Lemma 4.69. The restriction mapSIndUPσ → SIndOPσ (4.30)φ 7→ φ|Ois a continuous linear map and also a surjective homomorphism of TVS. Itmakes the following diagram commuteS(U, V ) S(O, V )SIndUPσ SIndOPσProof. Obviously the map is linear and the diagram commutes. The map(4.30) is continuous since its composition with S(U, V )→ SIndUPσ is contin-uous. It is a homomorphism of TVS since it is a surjective map to a Fre´chetspace hence always has closed image.4.6.4 Pseudo-Cosheaf Property of Schwartz InductionsSimilar to the Schwartz function spaces, the Schwartz induction spacesalso form a pseudo-cosheaf on certain open subsets of G.The Zariski P -Topology on GLet G,P be as the beginning of this section. The flag varietyP\G = P(R)\G(R) ' (P\G)(R)is a real projective algebraic variety, and the quotient morphismpi : G→ P\Gis continuous under Zariski topologies of G and P\G. We can pull back theZariski topology on P\G to have a subtopology of the Zariski topology onG.112Definition 4.70. A subset of the form pi−1(U) where U ⊂ P\G is a Zariskiopen subset of P\G, is called a Zariski P -open subset of G. Let T PG bethe family of all Zariski P -open subsets of G, and we call it the ZariskiP -topology on G.We have the following easy facts:Lemma 4.71. The T PG is a topology on G. In particular, the Zariski P -topology is quasi-compact.A Zariski open subset of G is Zariski P -open if and only if it is stableunder left P -translation.Example 4.72. Let PΩ = PΩ(R) be the real point group of a standardparabolic R-subgroup PΩ of G. For each w ∈ [WΘ\W/WΩ], the open subsetsGΩ≥w, GΩ>w are Zariski P -open subsets of G (see 3.3.3 for notations).Let NP be the opposite unipotent radical of P , then the PNP and itsright translations are Zariski P -open subsets of G.Pseudo-Cosheaf Property of Schwartz InductionLet U ∈ T PG be a Zariski P -open subset. We have the space of S(U, V ) ofSchwartz V -valued functions on U , and the local Schwartz induction spaceSIndUPσ on U . We also have the σ-mean value mapS(U, V )→ SIndUPσ, f 7→ fσwhich is a surjective homomorphism between NF-spaces.Let U1 ⊂ U2 be two Zariski P -open subsets, we have discussed the openextension map in Lemma 4.67:ExU2U1 : SIndU1P σ ↪→ SIndU2P σand seen it is an injective homomorphism of NF-spaces.With the local Schwartz induction spaces and extension maps definedabove, the correspondence U 7→ SIndUPσ is a pre-cosheaf on the Zariski P -topology of G. Moreover, by diagram-chasing, we see the coshear sequencesare exact, hence we haveLemma 4.73. The correspondence SInd−Pσ : U 7→ SIndUPσ is a pseudo-cosheaf of NF-spaces, on the Zariski P -topology of G.Remark 4.74. If the representation σ is a Nash-representation, the associ-ated vector bundle σ ×P G is a Nash bundle. The above cosheaf is exactlythe pull-back cosheaf of the pseudo-cosheaf of Schwartz sections of the as-sociated vector bundle.113Lemma 4.75. For each Zariski P -open subset U , the surjective homomor-phismS(U, V ) SIndUPσis functorial, hence we have a morphismS(−, V )→ SInd−Pσ (4.31)of cosheaves of NF-spaces on the Zariski P -topology, which is a surjectivehomomorphism on each Zariski P -open subset.4.6.5 Distributions on Schwartz Induction SpacesWe keep the setting as last subsection, and let F be an NF-space. Similarto the pseudo-sheaf of distributions on Schwartz functions, we define the F -valued distributions on the Schwartz induction spaces, and show they forma pseudo-sheaf on the Zariski P -topology on G.Definition of Distributions on Schwartz InductionsDefinition 4.76. Let L(SIndUPσ, F ) be the space of continuous linear mapsfrom SIndUPσ to F . The elements in it are called F -valued distributionson SIndUPσ.Since SIndUPσ is a NF-space, we haveLemma 4.77. The canonical map (SIndUPσ)′⊗F → L(SIndUPσ, F ) extendsto an isomorphism of TVS:(SIndUPσ)′ ⊗̂ F ∼−→ L(SIndUPσ, F ). (4.32)Pseudo-Sheaf Property of Distributions on Schwartz InductionsLet U1, U2 ∈ T PG be two open subsets in the Zariski P -topology of G,and U1 ⊂ U2. We have the extension map ExU2U1 : SIndU1P σ → SIndU2P σ, andletResU2U1 : L(SIndU2P σ, F )→ L(SIndU1P σ, F )D 7→ D ◦ ExU2U1be its F -transpose map. This map is a homomorphism of TVS, called therestriction map of distributions.114Remark 4.78. Keep in mind that we use the uppercase notation Ex,Resto denote the extensions of Schwartz inductions and restrictions of distribu-tions on Schwartz inductions, and lowercase notation ex, res to denote theextensions of Schwartz functions and restrictions of Schwartz distributions.The correspondenceL(SInd−Pσ, F ) : U 7→ L(SIndUPσ, F )(with the restriction maps Res) is a presheaf on the Zariski P -topology ofG. Moreover, we haveLemma 4.79. The correspondence L(SInd−Pσ, F ) : U 7→ L(SIndUPσ, F ) is apseudo-sheaf on the Zariski P -topology on G.Proof. By Lemma 4.77, we just need to show the case F = C. The pseudo-sheaf sequence is exact because it is the strong dual sequence of the pseudo-cosheaf sequence, which is exact by Lemma 4.73.Distributions Supported in Zariski P -Closed SubsetsBy the pseudo-sheaf property of L(SInd−Pσ, F ), we have the followingresults similar to the Lemma 4.38 and Lemma 4.39.Lemma 4.80. Let U1, U2 be two Zariski P -open subsets of G, such that U1 ⊂U2. Let Z be a nonsingular closed subvariety of U1 (hence also nonsingularsubvariety of G). Assume Z is stable under left P -translation. Then U1 −Z,U2−Z are Zariski P -open subsets of G, and one has the restriction mapsResU1U1−Z : L(SIndU1P σ, F )→ L(SIndU1−ZP σ, F )ResU2U2−Z : L(SIndU2P σ, F )→ L(SIndU2−ZP σ, F )and the restriction map ResU2U1 sends the kernelKer(ResU2U2−Z) isomorphicallyto the kernel Ker(ResU1U1−Z).Lemma 4.81. Let U1, U2 be two Zariski P -open subsets, and Z ⊂ G be anonsingular closed subvariety of both U1 and U2, and assume Z is stableunder left P -translation. Then the two kernel spaces Ker(ResU1U1−Z) andKer(ResU2U2−Z) are canonically isomorphic.The proof of the above two Lemmas are exactly the same as Lemma 4.38and Lemma 4.39.115Extensions of Distributions from Closed SubvarietiesLet U be a open subvariety of G, and O ⊂ U be a nonsingular closedsubvariety of U , assume both U and O are left P -stable. We have thesurjective homomorphism SIndUPσ → SIndOPσ by Lemma 4.69. The F -transpose of this map gives a homomorphism between the distribution spaceson Schwartz inductions, and we haveLemma 4.82. The F -transpose of the map SIndUPσ → SIndOPσ is an injec-tive homomorphism of TVS:L(SIndOPσ, F ) ↪→ L(SIndUPσ, F ) (4.33)Definition 4.83. We call the above map (4.33) the extension of distri-butions from L(SIndOPσ, F ) to L(SIndUPσ, F ).4.6.6 Group Actions on Local Schwartz InductionsSuppose H is a closed algebraic subgroup of G, and suppose the Y is leftP -stable and is an H-subvariety of G, i.e. it is a smooth subvariety of G,and the embedding Y → G is H-equivariant under the right H-translationson Y and G. Let SIndYP σ be the local Schwartz induction of σ from P toY .Definition 4.84 (Right regular action on local Schwartz inductions).Let φ ∈ SIndYP σ be an arbitrary element in the Schwartz induction. Leth ∈ H, and we define Rhφ by[Rhφ](y) := φ(yh), ∀y ∈ Y.This is a group action of H on the vector space SIndYP σ, called the rightregular H-action on SIndYP σ.We haveLemma 4.85. Under the right regular H-action, the SIndYP σ is a smoothH-representation. The map (4.28)S(Y, V ) SIndYP σis H-equivariant hence an intertwining operator between H-representations.116Group Actions are Compatible with Open ExtensionsLet U1, U2 be two Zariski P -open H-subvarieties of G, i.e. they are opensubvarieties stable under right H-translation and the embeddings are H-equivariant. Assume U1, U2, then as in Lemma 4.67, we have the extensionmap from SIndU1P σ to SIndU2P σ.Lemma 4.86. Under the right regular H-actions on SIndU1P σ and SIndU2P σ,the extension map (4.29)SIndU1P σ ↪→ SIndU2P σis an H-intertwining operator. Hence the following diagramS(U1, V ) S(U2, V )SIndU1P σ SIndU2P σis a diagram in the category of H-representations, with all four spaces en-dowed with the right regular H-actions.Group Actions are Compatible with Closed RestrictionsLet U be a Zariski P -open subvariety of G, O be a Zariski P -closed sub-variety of U . Assume they are all H-subvarieties of G. Let SIndUPσ,SIndOPσbe the Schwartz inductions on them respectively, and they have the rightregular H-actions, and the restriction map between them is H-equivariant.Lemma 4.87. Under the right regular H-actions, the restriction map (4.30)SIndUPσ SIndOPσis an H-intertwining operator. Then following diagramS(U, V ) S(O, V )SIndUPσ SIndOPσis in the category of H-representations, with all four spaces endowed withthe right regular H-actions.1174.6.7 Local Schwartz Inductions on FibrationsWe study the Schwartz induction spaces on subvarieties of G, which areisomorphic to direct products of P and subvarieties of G.In this subsection, we let P be the real point group of a parabolic R-subgroup, and let NP be the unipotent radical of its opposite parabolicsubgroup.Local Trivialization of pi : G→ P\GLet pi : G→ P\G be the algebraic quotient map. The (G,P\G, pi) is analgebraic fibre bundle, and it is locally trivial on each Zariski open subsetYw := pi(PNPw)of P\G. The inverse imageZw := pi−1(Yw) = PNPwis isomorphic to a direct product of varieties, i.e. the following map is anisomorphism of variety:P × w−1NPw ∼−→ Zw = PNPw(p, n) 7→ pwnDirect Product DecompositionLet Y ⊂ Yw be a nonsingular subvariety of Yw ⊂ P\G, and let Z =pi−1(Y ). We assume there is a subvariety O of G, and the multiplicationmap on G induces an isomorphism of varieties:P ×O ∼−→ Z = pi−1(Y )(p, x) 7→ pxThen this isomorphism is a P -equivariant isomorphism, with P ×O and Zendowed with the left P -translations. The O is isomorphic to the subvariety{e} ×O of Z.Example 4.88. We will only consider the following two concrete examplesin this thesis:• Let Y = Yw (and Z = Zw as above), the O is exactly the w−1NPw.118• Let Y = pi(PwP∅), it is the P∅-orbit on the P\G through the pointxw = pi(Pw). Then Z = PwP∅, and the O is the subgroup w−1NPw∩N∅.The isomorphism P × O ∼−→ Z induces the following isomorphisms ofTVS:S(P, V ) ⊗̂ S(O,C) ∼−→ S(Z, V )S(P,C) ⊗̂ S(O, V ) ∼−→ S(Z, V )(This follows from E-6 of Proposition 4.30.) We know the algebraic tensorproducts S(P, V )⊗S(O,C) is dense in S(P, V )⊗̂S(O,C), and the S(P,C)⊗S(O, V ) is dense in S(P,C)⊗̂S(O, V ). We first study the images of functionsin these algebraic tensor products, under the σ-mean value map.Lemma 4.89. Let φ ∈ S(P, V ) and ψ ∈ S(O,C), and letφ⊗ ψ : Z ' P ×O → V(p, x) 7→ ψ(x)φ(p)Then φ⊗ ψ ∈ S(Z, V ), and its image under the map S(Z, V )→ SIndZPσ is(φ⊗ ψ)σ(px) = ψ(x)φσ(p) (4.34)Proof. This is easy to verify:(φ⊗ ψ)σ(px) =∫Pσ(q−1)(φ⊗ ψ)(q · px)dq=∫Pσ(q−1)(φ⊗ ψ)(qpx)dq=∫Pσ(q−1)[ψ(x)φ(qp)]dq= ψ(x)∫Pσ(q−1)φ(qp)dq= ψ(x)φσ(p)Lemma 4.90. Let φ ∈ S(P,C) and ψ ∈ S(O, V ), and letφ⊗ ψ : Z = P ×O → V(p, x) 7→ φ(p)ψ(x)119Then φ⊗ ψ ∈ S(Z, V ), and its image under map S(Z, V )→ SIndZPσ is(φ⊗ ψ)σ(px) = σ(p)∫Pφ(q)σ(q−1)ψ(x)dq (4.35)Proof. Actually(φ⊗ ψ)σ(px) =∫Pσ(q−1)(φ⊗ ψ)(qpx)dq=∫Pφ(qp)σ(q−1)ψ(x)dq=∫Pφ(qp)σ(p)σ(qp)−1ψ(x)dq= σ(p)∫Pφ(qp)σ(qp)−1ψ(x)dq= σ(p)∫Pφ(qp)σ(qp)−1ψ(x)dqp= σ(p)∫Pφ(q)σ(q)−1ψ(x)dqThe First IsomorphismBy Lemma 4.89, we have the following mapSIndPPσ ⊗ S(O,C)→ SIndZPσφσ ⊗ ψ 7→ (φ⊗ ψ)σIt is easy to see this map is well-defined, and is independent of the choiceof φ ∈ S(P, V ). Actually let φ0 ∈ Ker{S(P, V ) → SIndPPσ}, then for anyψ ∈ S(O,C), the φ0 ⊗ ψ is in the kernel of map S(Z, V ) → SIndZPσ byLemma 4.89. Moreover, we haveLemma 4.91. The above map φσ⊗ψ 7→ (φ⊗ψ)σ extends to an isomorphismon the completion:SIndPPσ ⊗̂ S(O,C) ∼−→ SIndZPσ (4.36)and the following diagram commutes:120S(P, V ) ⊗̂ S(O,C) S(Z, V )SIndPPσ ⊗̂ S(O,C) SIndZPσ'(4.36)Proof. By Lemma 4.89, the diagram commutes, hence the (4.36) is surjec-tive. If (φ⊗ ψ)σ = 0, then by Lemma 4.89 again, we see ψ(x)φσ(p) = 0 forall p ∈ P, x ∈ O. This is means either φσ = 0 or ψ = 0, otherwise there existp ∈ P, x ∈ O such that φσ(p) 6= 0, ψ(x) 6= 0 thus ψ(x)φσ(p) 6= 0. Hence themap φσ ⊗ ψ 7→ (φ⊗ ψ)σ is injective, so is the map (4.36).The Second IsomorphismFor a F ∈ SIndZPσ, it is a smooth function on Z. Let F |O be its restric-tion to the submanifold O (remember O is embedded into Z as {e} × O),then F |O is a smooth V -valued function on O.Lemma 4.92. We have(1) The F |O is a Schwartz function on O, i.e. F |O ∈ S(O, V ).(2) The restriction mapSIndZPσ → S(O, V ) (4.37)F 7→ F |Ois a homomorphism of TVS.(3) The restriction map (4.37) is an isomorphism between TVS.Proof. (1) We first consider F of the form (φ⊗ ψ)σ where φ ∈ S(P, V ) andψ ∈ S(O,C). By Lemma 4.89, we haveF (p, x) = ψ(x)φσ(p), ∀p ∈ P, x ∈ O.Hence F |O(x) = F (e, x) = ψ(x)φσ(e). This is a scalar Schwartz function,since φσ(e) is a fixed vector in V , and ψ is a scalar valued Schwartz function.Hence for F = (φ⊗ ψ)σ, we have F |O ∈ S(O, V ).Since the S(O, V ) is complete, and functions of the form (φ ⊗ ψ)σ aredense in SIndZPσ, we see the map F 7→ F |O has its image in S(O, V ).(2) All spaces are nuclear, hence we just need to show (4.37) is continu-ous. Actually it is continuous since its composition with S(Z, V )→ SIndZPσis continuous on the dense subspace S(P, V )⊗ S(O,C).121(3) We show the (4.37) is an isomorphism. Obviously it is injective, sincea function in SIndZPσ is uniquely determined by its values on O ' {e} ×O.We just need to show it is surjective.For any Ψ ∈ S(O, V ), let γ ∈ C∞c (P,C) be a bump function satisfying∫Pγ(p)dp = γ(e) = 1.We construct a function F on Z = P ×O byF (p, x) := γ(p)σ(p)Ψ(x), ∀p ∈ P, x ∈ O.This F is in S(Z, V ) since it is smooth with compact support. It is easy toverifyF σ(p, x) =∫Pσ(q−1)F (qp, x)dq=∫Pσ(q−1)γ(qp)σ(qp)Ψ(x)dq=∫Pγ(qp)σ(p)Ψ(x)dq= [∫Pγ(qp)dq] · σ(p)Ψ(x)= [∫Pγ(q)dq] · σ(p)Ψ(x)= σ(p)Ψ(x)Then F σ|O = Ψ. Hence (4.37) is surjective.The Trivial Case SIndPPσIf P = G, a function in SIndPPσ is uniquely determined by its value atidentity e:Lemma 4.93. The delta functionΩe : SIndPPσ → Vφ 7→ φ(e)is an isomorphism of TVS.Proof. This is the special case of the above Lemma. Since P = G, thequotient variety P\G = {∗} is a singleton. Then Y = {∗}, Z = G, andO = {e}. The S(O, V ) = V , and the map SIndPPσ → S(O, V ), F 7→ F |O isexactly the delta function Ωe.122Chapter 5Intertwining DistributionsSummary of This ChapterThis is a conceptual chapter, in which we apply the tools of Schwartzanalysis developed in the last chapter, to study intertwining operators. Weembed the space HomG(I, J) of intertwining operators between two smoothparabolic inductions I, J , to the space of equivariant Schwartz distributionson the Schwartz induction spaces, and study such distributions by lookingat their restrictions to various Zariski open subsets (unions of double cosets)of G.In the first three sections, we will work on more general situation, sinceour long term goal is to apply the theory to study all intertwining operatorsbetween two arbitrary parabolic inductions. The main theme of the thesis isthe irreducibility of a single parabolic induction, and in the last two sectionswe will look at the space of self-intertwining operators.Part I—General Intertwining DistributionsIn section 5.1, 5.2, 5.3, we work in general situation, and letG = a connected reductive linear algebraic group defined over RP,Q = two parabolic R-subgroups of GG,P,Q = the corresponding Lie groups of real points of G,P,Q(σ1, V1) = a nuclear Harish-Chandra representation of P(σ2, V2) = a nuclear Harish-Chandra representation of QI = C∞IndGPσ1 = SIndGPσ1J = C∞IndGQσ2 = SIndGQσ2For two TVS E1, E2, letL(E1, E2) = Homcont(E1, E2)= the space of continuous linear maps from E1 to E2.123We study the space HomG(I, J) of intertwining operators between the twosmooth inductions (Schwartz inductions) I and J .In 5.1, we embed the space HomG(I, J) of intertwining operators, intothe space L(I, V2) ⊂ L(S(G,V1), V2) of V2-valued Schwartz distributions onSIndGPσ1 (or S(G,V1)), and starting from this section we call elements inHomG(I, J) intertwining distributions.In 5.2, we show the intertwining distributions in HomG(I, J) have sup-ports equal to closed unions of (P,Q)-double cosets. Since there are finitelymany double cosets on G, we find the first discreteness of intertwining op-erators: they are sorted into finitely many families with different supports.(One can also replace the Q by its algebraic subgroups H such that G hasfinitely many (P,H)-double cosets.)In 5.3, we first introduce the “maximal double coset” in the support.If an intertwining distribution D has “a” maximal double coset Gw in itssupport, then the restriction of D to the open subset G≥w is a nonzeroelement which vanishes on the smaller open subset G>w, i.e. this restrictionis a nonzero element in the kernel of the restriction map from G≥w to G>w.Part II—Self-Intertwining Distributions and IrreducibilitiesIn section 5.4, 5.5, we work on the self-intertwining distributions, andlet• G (resp. S,P∅) be a connected reductive linear algebraic group definedover R (resp. a maximal R-split torus, a minimal parabolic R-subgroupcontaining S).• P = PΘ be a standard parabolic R-subgroup containing P∅ (corre-sponding to a subset Θ of the base). In this chapter we will not usethe particular set Θ and we drop the subscript Θ since there is noambiguity.• MP = MΘ,NP = NΘ be the standard R-Levi factor and unipotentradical of P, M∅,N∅ be the Levi R-factor and unipotent radical of P∅;• G,P∅,M∅, N∅, P,MP , NP etc. be the Lie groups of R-rational pointscorresponding to the above algebraic groups denoted by boldface let-ters.• (σ, V ) be a nuclear Harish-Chandra representation of P .• I = C∞IndGPσ = SIndGPσ be the smooth (Schwartz) induction.124We apply the notions and results in Part I to the case when P = Q, σ1 =σ2, I = J , and study the space HomG(I, I) and the irreducibility of thesmooth induction I.Section 5.4 now consists of some notations and basic settings. Also werecall the fact that the intertwining distributions corresponding to scalarintertwining operators are supported in suppD (see Lemma 5.21).In 5.5, we give the local descriptions of intertwining distributions basedon the maximal double cosets in their supports. We summarize two mainsteps to show the HomG(I, I) = C.5.1 Intertwining DistributionsIn this and the following two sections, we let G and G = G(R) bethe same as the beginning of this chapter. Let P,Q be two parabolic R-subgroups of G, and let P = P(R), Q = Q(R) be their corresponding Liegroups of real points. Let(σ1, V1) = a Harish-Chandra representation of P(σ2, V2) = a Harish-Chandra representation of QIn particular, the V1, V2 are nuclear Fre´chet spaces and smooth representa-tions of P,Q respectively.Let SIndGPσ1,SIndGQσ2 be the Schwartz inductions of σ1, σ2 respectively.Since the quotient manifolds P\G,Q\G are compact, the Schwartz induc-tions agree with the smooth inductions. For simplicity, we use the followingnotations for the Schwartz (smooth) inductions:I = C∞IndGPσ1 = SIndGPσ1J = C∞IndGQσ2 = SIndGQσ25.1.1 Intertwining DistributionsAs in 4.6.1, for a Schwartz V1-valued function f ∈ S(G,V1), its σ1-meanvalue functionfσ1(g) =∫Pσ1(p−1)f(pg)dp, g ∈ Gis in the Schwartz induction space SIndGPσ1, and the map S(G,V1) →SIndGPσ1, f 7→ fσ1 is a surjective homomorphism of TVS. And Frobeniusreciprocity (Lemma 2.31) tells us the map T 7→ Ωe ◦ T is an isomorphismbetween (finite dimensional) vector spaces.125Given an intertwining operator T ∈ HomG(I, J), we consider the com-position map DT : I → V2 of the three maps: (1) the mean value mapS(G,V1) → I, f 7→ fσ1 ; (2) the operator T : I → J ; (3) the delta functionΩe : J → V2:S(G,V1) V2I JDTTΩeThe DT is a V2-valued Schwartz V1-distribution on G, and it is obviouslyQ-equivariant, since Ωe is Q-equivariant, T and f 7→ fσ1 are G-equivariant.Since the σ1-mean value map S(G,V1)→ I is surjective, its adjoint mapis injective, and the correspondence T 7→ Ωe ◦ T is bijective, hence we have:Lemma 5.1. The following linear mapHomG(I, J)→ HomQ(S(G,V1), V2) ⊂ L(S(G,V1), V2) (5.1)T 7→ DTis one-to-one.Thus we have embedded the finite dimensional space HomG(I, J), as asubspace of L(S(G,V1), V2). From now on, we identify the space HomG(I, J)with its image in the L(S(G,V1), V2), and identify an intertwining operatorT with its corresponding distribution DT .Definition 5.2. Let D ∈ HomG(I, J).• We call the DT ∈ L(S(G,V1), V2) a V2-valued intertwining distri-bution on S(G,V1).• We call the Ωe◦T ∈ L(I, V2) a V2-valued intertwining distributionon I.Remark 5.3. The above term “intertwining distribution” is an emphasisof the fact that a distribution come from an intertwining operator.• When the spaces under discussion are clear without ambiguity, we willsimply use the term intertwining distributions, without mentioningthe spaces S(G,V1), I, V2 etc.• In application we will also study the restrictions of intertwining distri-butions on S(G,V1) or I to their subspaces, and we will also call theserestricted distributions intertwining distributions (abuse of terms).126• By abuse of terms, we will also call elements in HomG(I, J) inter-twining distributions, instead of intertwining operators. The space ofintertwining distributions is exactly HomG(I, J).The finiteness of HomG(I, J) indicates that there should not be manyintertwining distributions, hence there should be a lot of restrictions onsuch distributions. In the next section, we will see the first good property ofintertwining distributions on S(G,V1), i.e. their supports are good subsetsof G.5.2 Supports of Intertwining DistributionsWe keep the setting as in 5.1, and let D ∈ HomG(I, J) ⊂ L(S(G,V1), V2)be an intertwining distribution. As we have seen in section 4.4, the D has awell-defined support under the Zariski topology of G. Let suppD and vanDbe the support and maximal vanishing subset as in Definition 4.48. In thissection, we showLemma 5.4. The support suppD and maximal vanishing subset vanD ofD are stable under left P -translation and right Q-translation. In particular,the suppD is a (Zariski closed) union of (P,Q)-double cosets in G, the vanDis a (Zariski open) union of (P,Q)-double cosets in G.Remark 5.5. Let P′ ⊂ P and Q′ ⊂ Q be two parabolic R-subgroupscontained in P,Q respectively, and let P ′, Q′ be the corresponding Lie sub-groups of R-rational points. Then the suppD and vanD are also unions of(P ′, Q′)-double cosets in G.Before studying the supports, we have the following trivial Lemma:Lemma 5.6. Let D be an intertwining distribution. Then the followingstatements are equivalent:• D = 0 or equivalently the corresponding intertwining operator is zero.• suppD = ∅.5.2.1 Some Topological FactsLet H ⊂ G be an algebraic subgroup, i.e. there is an R-closed subgroupH of G such that H = H(R).127Definition 5.7. For a Zariski open subset U ⊂ G, and an element g ∈ G,let Ug be the right translation of U by g, and gU be the left translation ofU by g. We call theUH :=⋃h∈HUhthe right H-augmentation of U , and similarly call theHU :=⋃h∈HhUthe left H-augmentation of U .We have the following basic facts:Lemma 5.8. Let U ⊂ G be a Zariski open subset.1. For a g ∈ G, the Ug and gU are Zariski open in G.2. For an algebraic subgroup H ⊂ G, the UH and HU are Zariski opensubsets of G, since they are union of translations of U . Actually, theUH is a finite union of Zariski open subsets of the form Uh for someh ∈ H, and HU is a finite union of open subsets of the form hU forsome h ∈ H.3. A subset S of G is left H-stable (i.e. hS = S for all h ∈ H), if andonly if HS = S; the S is right H-stable (i.e. Sh = S for all h ∈ H),if and only if SH = S.Proof. Part 1 and 2 are trivial. We show part 3 for the “left part”, the“right part” is similar.Suppose S is left H-stable, i.e. ∀h ∈ H, we have hS = S. Then hS ⊂ Sfor all h ∈ H, hence their union HS is contained in S, and S ⊂ HS isobvious. Hence HS = S.Suppose HS = S, then for all h ∈ H, hS ⊂ HS ⊂ S. By the sameargument h−1S ⊂ S, and translate both sides by h we have S ⊂ hS for allh ∈ H. Hence for all h, we have hS = S.5.2.2 The suppD and vanD are Right Q-StableLet D be an intertwining distribution. We show the vanD is stable underright Q-translation, which implies suppD is also right Q-stable since it isthe complement of vanD. More precisely, by part 3 of the above Lemma,we need to show(vanD)Q = vanD.128The inclusion vanD ⊂ (vanD)Q is obvious, hence we just need to show(vanD)Q ⊂ vanD, or equivalently (vanD)q ⊂ vanD for all q ∈ Q. By themaximality of vanD, we just need to show D vanishes on (vanD)q. ByLemma 4.49, we know D vanishes on vanD, and we just need to apply thefollowing Lemma to U = vanD:Lemma 5.9. Let U be a Zariski open subset of G, and D|U = 0. ThenD|Uq = 0 for all q ∈ Q.Proof. Suppose the intertwining distribution D ∈ HomQ(S(G,V1), V2) van-ishes on U , namely 〈D,φ〉 = 0 for all φ ∈ S(U, V1) ⊂ S(G,V1). We show〈D, f〉 = 0 for all f ∈ S(Uq, V1) ⊂ S(G,V1).Actuallyf is in the subspace S(Uq, V1) ⊂ S(G,V1)⇔f vanishes with all derivatives on G− Uq⇔Rqf vanishes with all derivatives on G− U⇔Rqf ∈ S(U, V1) ⊂ S(G,V1)Hence if f ∈ S(Uq, V1), we know the right translation Rqf is in S(U, V1)and by the assumption on D, we have 〈D,Rqf〉 = 0. Now we have〈D, f〉 = 〈D,Rq−1Rqf〉= σ2(q−1)〈D,Rqf〉 (D is Q-equivariant)= σ2(q−1)0Thus we have shown 〈D, f〉 = 0 for all f ∈ S(Uq, V1), hence D vanishes onUq.5.2.3 The suppD and vanD are Left P -StableLet D be an intertwining distribution as above. We show the vanDis left P -stable, which implies suppD is also left P -stable. Similar to theabove discussion, we just need to show P (vanD) ⊂ vanD, or equivalentlyp(vanD) ⊂ vanD for all p ∈ P .Lemma 5.10. Let f ∈ S(G,V1) be and arbitrary Schwartz V1-valued func-tion, and p ∈ P be an arbitrary element. Let(Lpf)(g) := f(p−1g),∀g ∈ G129be the left translation of f by p. Let σ1(p)f be the following compositionfunction[σ1(p)f ](g) := σ1(p)f(g),i.e. σ1(p) acts on the vector f(g) ∈ V1. Then we have(1) The Lpf is in S(G,V1).(2) The σ1(p)f is in S(G,V1).(3) The σ1-mean value function of Lpf (the image of Lpf in SIndGPσ1) is(Lpf)σ1 = δP (p)[σ1(p−1)f ]σ1 ,i.e. it is the σ1-mean value function of σ1(p−1)f multiplied by a constantδP (p).Proof. Part (1) is true because the left translation by p is an algebraicisomorphism on G, and it induces an isomorphism on Schwartz functionspaces:Lp : S(G,V1) ∼−→ S(G,V1).Part (2) is true because the function σ1(p)f is simply a composition ofthe Schwartz function f with a single linear operator σ1(p).We verify part (3), let f ∈ S(G,V1) and p ∈ P be arbitrary elements.Then(Lpf)σ1(g) =∫Pσ1(q−1)[Lpf ](qg)dq=∫Pσ1(q−1)f(p−1qg)dq=∫Pσ1(q−1)σ1(p)σ1(p−1)f(p−1qg)dq=∫Pσ1(p−1q)−1[σ1(p−1)f ](p−1qg)dq=∫Pσ1(p′)−1[σ1(p−1)f ](p′g)d(pp′) (p′ = p−1q)= δP (p)∫Pσ1(p′)−1[σ1(p−1)f ](p′g)dp′= δP (p)[σ1(p−1)f ]σ1(g)hence (Lpf)σ1 = δP (p)[σ1(p−1)f ]σ1 .130Similar to the proof of right Q-stability of vanD, we show the followingLemma which implies vanD is left P -stable:Lemma 5.11. Suppose U is a Zariski open subset on which D vanishes(D|U = 0), then for an arbitrary p ∈ P , we have D|pU = 0, namely D alsovanishes on the left translation pU .Proof. We just need to show 〈D, f〉 = 0 for all f ∈ S(pU, V1). Supposethe intertwining distribution D come from an intertwining operator T ∈HomG(I, J).First the isomorphismLp−1 : S(G,V1) ∼−→ S(G,V1)maps the subspace S(pU, V1) of S(G,V1) isomorphically to the subspaceS(U, V1), i.e. f ∈ S(pU, V1) is equivalent to Lp−1f ∈ S(U, V1). Then for allf ∈ S(pU, V1)〈D, f〉 = 〈D,LpLp−1f〉= 〈Ωe ◦ T, [LpLp−1f ]σ1〉= 〈Ωe ◦ T, δP (p)[σ1(p−1)Lp−1f ]σ1〉= δP (p)〈Ωe ◦ T, [σ1(p−1)Lp−1f ]σ1〉= δP (p)〈D,σ1(p−1)Lp−1f〉= δP (p) · 0The last equality holds because Lp−1f is in S(U, V1) and the compositionfunction σ1(p−1)Lp−1f is also in S(U, V1), since it vanishes with all deriva-tives on G−U . By the assumption D|U = 0, we know 〈D,σ1(p−1)Lp−1f〉 =0.Therefore we have shown 〈D, f〉 = 0 for all f ∈ S(pU, V1), hence Dvanishes on pU .Remark 5.12. We summarize the above results we have proved for thesupport of intertwining distributions. We first note the following sequenceof inclusions:HomG(I, J) = HomQ(I, V2) ⊂ HomQ(S(G,V1), V2) ⊂ L(S(G,V1), V2).A distribution D in the largest space L(S(G,V1), V2) has a well-definedsupport as in Chapter 4. IfD is in the subspace HomQ(S(G,V1), V2), then itssupport is right Q-stable. If further D is in the HomG(I, J) = HomQ(I, V2),i.e. it factor through the S(G,V1)→ I, then its support is also left P -stable.1315.3 Some Notions of Distribution Analysis on GLet G,P,Q, (σ1, V1), (σ2, V2), I, J be the same as in 5.1. In this section,we introduce the notions of maximal double coset(s) in the supports, andthe diagonal actions on Schwartz distribution spaces. Each subsection couldbe read independently.5.3.1 Maximal Double Cosets in SupportsLet D ∈ HomG(I, J) be an intertwining distribution G, then the suppDis a closed union of (P,Q)-double cosets. Let H ⊂ Q be an algebraic sub-group, i.e. assume there is a R-closed subgroup H ⊂ Q such that H = H(R).In this subsection, we assume there are finitely many (P,H)-double coseton G, and they are parameterized by a finite set W.Since there are finitely many (P,H)-double cosets, as in 3.3.3, the doublecosets form a partial ordered set, ordered by their Zariski closures. And wecan find a maximal (P,H)-double coset from the suppD if suppD 6= ∅ (orequivalently D 6= 0).Remark 5.13. Note that the closure order on double cosets is only a partialorder. If a double coset is maximal in suppD, it doesn’t mean it is “greater”than all other double cosets in suppD, but means it is “not smaller” thanany other double cosets in suppD.We call it “a”, but not “the” maximal double coset, because there mightbe more than one maximal double cosets in suppD since the order is only apartial order.For a w ∈ W, we denote the corresponding (P,H)-double coset by Gw.We can define the Zariski open subsets G≥w, G>w as in 3.3.3, and Gw =G≥w −G>w is closed in G≥w. We denote byresGG≥w = the restriction map L(S(G,V1), V2)→ L(S(G≥w, V1), V2)resGG>w = the restriction map L(S(G,V1), V2)→ L(S(G>w, V1), V2)The following easy lemma gives an algebraic description of “a” maximaldouble coset in suppD:Lemma 5.14. Let D ∈ HomQ(S(G,V1), V2) be an intertwining distributionand assume D 6= 0 to make sure it has nonempty support, then the following3 statements are equivalent:1. The Gw is “a” maximal double coset in suppD.1322. The restriction D|G≥w = resGG≥w(D) 6= 0 and the restriction D|G>w =resGG>w(D) = 0.3. The D ∈ Ker(resGG>w) but D /∈ Ker(resGG≥w).Proof. We show 1.⇔ 2. since 2.⇔ 3. is trivial.(1.⇒ 2.) Suppose Gw is a maximal double coset in suppD.First we have Gw ⊂ suppD, hence G≥w ∩ suppD ⊃ Gw 6= ∅. HenceG≥w * vanD and D|G≥w 6= 0.Second if D|G>w 6= 0, then G>w * vanD hence G>w∩suppD 6= ∅. Thereexists a Gx ⊂ G>w ∩ suppD (thus Gx > Gw). Hence Gw is not maximal,contradiction!(2. ⇒ 1.) Suppose D|G≥w 6= 0 and D|G>w = 0. First we have G>w ⊂vanD.Second we show Gw ⊂ suppD. Suppose not, then Gw ⊂ vanD, thenG≥w = Gw ∪G>w ⊂ vanD, and D|G≥w = 0, contradiction!Now we see the Gw is a double coset in suppD, and for all x > w, theGx ⊂ G>w ⊂ vanD, i.e. all orbits “greater” than Gw are not contained insuppD. Hence Gw is “a” maximal orbit in suppD.Remark 5.15. For D 6= 0, since there is at least one maximal doublecoset in suppD, the subsets Ker(resGG>w) − Ker(resGG≥w) “cover” the subsetHomG(I, J)− {0} of L(S(G,V1), V2). However this is not a partition, sincemaximal double coset(s) may not be unique.5.3.2 Diagonal Actions on Distribution Spaces—ILet P,Q, (σ1, V1), (σ2, V2) be as the beginning of this section. As in 4.5.5,letH = a closed algebraic subgroup of QY = a real subvariety of G which is closed under right H-translationOne has the right regular H-action on the Schwartz function space S(Y, V1),and the H-action on V2 through the representation σ2. (Note that H ⊂ Qis a subgroup).Definition 5.16 (Diagonal H-action on distribution space). We definethe following H-action L(S(Y, V1), V2): for all Φ ∈ L(S(Y, V1), V2), h ∈ Hand f ∈ S(Y, V1), the h · Φ is given by〈h · Φ, f〉 := σ2(h)〈Φ, Rh−1f〉. (5.2)133Here the〈, 〉 : L(S(Y, V1), V2)× S(Y, V1)→ V2is the pairing between the Schwartz distributions and Schwartz functions,Rh−1f is the right regular action of h−1 on f , the 〈Φ, Rh−1f〉 is a vector inV2, and h acts on it through the representation σ2.We call this action the diagonal H-action on the distribution spaceL(S(Y, V1), V2).Lemma 5.17. For the above diagonal H-action on L(S(Y, V1), V2), we have:1. The L(S(Y, V1), V2) is a smooth H-representation. It is isomorphic tothe tensor product representationL(S(Y, V1), V2) ' S(Y, V1)′ ⊗̂ V2 (5.3)of H, where S(Y, V1)′ is endowed with the contragredient H-action ofthe above right regular H-action, and V2 is endowed with the restrictedrepresentation σ2|H of H.2. The space HomH(S(Y, V1), V2) of H-equivariant V2-valued SchwartzV1-distributions on Y is exactly the space of H-invariants on the spaceL(S(Y, V1), V2):HomH(S(Y, V1), V2) = H0(H,L(S(Y, V1), V2)) (5.4)3. Let U1, U2 be two right H-stable Zariski open subvarieties of G andU1 ⊂ U2, the restriction map of distributionsresU2U1 : L(S(U2, V1), V2)→ L(S(U1, V1), V2)is an H-intertwining operator. Its kernel is a H-subrepresentation ofL(S(U2, V1), V2) under the diagonal H-action.4. Let U be a right H-stable Zariski open subvariety of G, O ⊂ U be aright H-stable Zariski closed subvariety of U . Then the inclusion map(extension of distributions from a closed subset)L(S(O, V1), V2) ↪→ L(S(U, V1), V2)is an H-intertwining operator.1345.3.3 Diagonal Actions on Distribution Spaces—IILet P,Q, (σ1, V1), (σ2, V2) be as above. Similar to the diagonal actionon Schwartz distributions on Schwartz function spaces, we can define thediagonal action on Schwartz distributions on Schwartz induction spaces.LetH = a closed algebraic subgroup of QY = a real subvariety of G which is stable under left P -translationand right H-translationSince Y is left a P -stable subvariety of G, one can define the local Schwartzinduction spaceSIndYP σ1as in 4.6. This space is a smooth H-representation, under the right regularH-action on it. The H also acts on the V2 through σ2 since it is a subgroupof Q.Definition 5.18 (Diagonal H-action on Schwartz induction spaces).Let L(SIndYP σ1, V2) be the space of V2-valued Schwartz distributions onSIndYP σ1. We define the following H-action on it: for each h ∈ H,Φ ∈L(SIndYP σ1, V2), φ ∈ SIndYP σ1, let h · Φ be the distribution〈h · Φ, φ〉 := σ(h)〈Φ, Rh−1φ〉 (5.5)Here Rh−1φ is the right regular action of h−1 on φ ∈ SIndYP σ1, the〈, 〉 : L(SIndYP σ1, V2)× SIndYP σ1 → V2is the pairing between Schwartz induction space SIndYP σ1 and Schwartzdistribution space on it, and 〈Φ, Rh−1φ〉 is a vector in V2 and σ2(h) actson it.We call this H-action the diagonal H-action on the distributionspace L(SIndYP σ1, V2).Lemma 5.19. For the above diagonal H-action, we have1. The L(SIndYP σ1, V2) is a smooth H-representation under the diagonalH-action. It is isomorphic to the tensor product H-representationL(SIndYP σ1, V2) ' (SIndYP σ1)′ ⊗̂ V2 (5.6)where the (SIndYP σ1)′ is the contragredient H-representation of theright regular representation, V2 is regarded as the representation spaceof restricted representation σ2|H .1352. The space HomH(SIndYP σ1, V2) of H-intertwining operators betweenSIndYP σ1 and (σ2|H , V2) is exactly the space of H-invariant of the spaceL(SIndYP σ1, V2):HomH(SIndYP σ1, V2) = H0(H,L(SIndYP σ1, V2)).3. Let U1, U2 be two Zariski open subvarieties of G, stable under left P -translation and right H-translation, and assume U1 ⊂ U2. Then therestriction map of distributionResU2U1 : L(SIndU2P σ1, V2)→ L(SIndU1P σ1, V2)is an H-intertwining operator under the diagonal H-actions. Its kernelis an H-subrepresentation of L(SIndU2P σ1, V2).4. Let U be a Zariski open subvariety of G, O be a Zariski closed subvari-ety of U (thus also a locally closed subvariety of G), and assume bothU and O are stable under left P -translation and right H-translation.Then the inclusion map of distributionsL(SIndOPσ1, V2) ↪→ L(SIndUPσ1, V2)is an H-intertwining operator.5.4 Self-Intertwining DistributionsStarting from this section through the entire chapter, we consider theself-intertwining distributions (operators) on a single smooth parabolic in-duction. We letP = PΘ = a standard parabolic R-subgroupcorresponding to a subset Θ of the base ∆P = PΘ = P(R)(σ, V ) = a Harish-Chandra representation of PI = SIndGPσ = C∞IndGPσWe apply the notions and results in the previous three sections, to the casewhenP = Q = PΘP = Q = PΘ(σ1, V1) = (σ2, V2) = (σ, V )I = J = SIndGPσ136i.e. P,Q are the same parabolic subgroup PΘ, and (σ1, V1), (σ2, V2) are thesame representation (σ, V ), I, J are the same Schwartz induction. And westudy the space HomG(I, I) of self-intertwining distributions (opera-tors).Let D ∈ HomG(I, I) = HomP (I, V ) ⊂ HomP (S(G,V ), V ) be an inter-twining distribution. As we have seen in 5.2, its support suppD is a union of(P, P )-double cosets of G. Actually for any algebraic subgroup H ⊂ P , thesupport suppD is also a union of (P,H)-double cosets. We are particularlyinterested in the case whenH = PΩ,where Ω is a subset of Θ, and PΩ is the standard real parabolic subgroupcorresponds to Ω.5.4.1 (P, PΩ)-Stable Subsets of G and Local SchwartzInductionsAs above, letΩ = a subset of ΘPΩ = the standard parabolic R-subgroup corresponding to ΩPΩ = PΩ(R) = the Lie group of real points of PΩThen by ∅ ⊂ Ω ⊂ Θ, we have P∅ ⊂ PΩ ⊂ PΘ.Notations on Double CosetsAs in 3.3, the (P, PΩ)-double cosets in G are parameterized by the set[WΘ\W/WΩ] = {w ∈W : w−1Θ > 0, wΩ > 0}of minimal representatives of the double quotient WΘ\W/WΩ. The doublecosets form a partial ordered set by the closure order on them, which alsogive the parameter set [WΘ\W/WΩ] a partial order. We adopt the same137notations as in 3.3:GΩw = PwPΩ = the double coset corresponding to wGΩ≥w =∐x≥wGΩx= the open union of double cosets PxPΩ s.t. PwPΩ ⊂ PxPΩGΩ>w =∐x≥w,x6=wGΩx= the open complement of GΩw in GΩ≥wIn particular, if Ω = ∅ (the empty set), we omit the superscript Ω, i.e.for w ∈ [WΘ\W ] = [WΘ\W/W∅], we use the following simplified notations:Gw = G∅w = PwP∅G≥w = G∅≥wG>w = G∅>wRemark 5.20. Note that for different Ω ⊂ Θ, the double cosets are param-eterized by different sets of minimal representatives [WΘ\W/WΩ]. Howeverall of them contain the minimal element e (identity of W ). And we alwayshaveGΩe = PGΩ≥e = GGΩ>e = G− Pno matter what Ω is, and they are stable under left and right P -translations.In this sense, the minimal double coset GΩe = P will be singled out and studiedseparately.Some Notations on Local Schwartz InductionsLet Y = GΩw, GΩ≥w, or GΩ>w. Obviously they are left P -stable and rightPΩ-stable, and the local Schwartz inductions SIndYP σ are smooth nuclearrepresentations of PΩ under the right regular PΩ-actions. We use the follow-ing simplified notations to denote the local Schwartz inductions on them:IΩw = SIndGΩwP σIΩ≥w = SIndGΩ≥wP σIΩ>w = SIndGΩ>wP σ138They are nuclear Fre´chet spaces and smooth representations of PΩ underthe right regular PΩ-actions.As before, when Ω = ∅ (empty set), we omit the superscript Ω, and foreach w ∈ [WΘ\W ], we letIw = I∅wI≥w = I∅≥wI>w = I∅>w5.4.2 Scalar Intertwining OperatorsLet G,P, σ, I be as above. The space HomG(I, I) always contain the1-dimensional subspace of scalar intertwining operators.Given a λ ∈ C, let λid ∈ HomG(I, I) be the scalar intertwining operator.Then the corresponding intertwining distribution (by Frobenius reciprocity)is obviously theλΩe : I → V.We call this λΩe ∈ HomP (I, I) a scalar intertwining distribution. Forsuch distributions, we obviously haveLemma 5.21. A scalar intertwining distribution λΩe ∈ HomP (I, V ) haveits support contained in P . More precisely, we havesupp(λΩe) ={P, if λ 6= 0∅, otherwise. (5.7)5.4.3 Analysis of Schwartz DistributionsLet Y be a subvariety of G, which is stable under left P -translation andright PΩ-translation. We have the Schwartz function space S(Y, V ) and theSchwartz induction space SIndYP σ, and a surjective homomorphism of TVSS(Y, V ) SIndYP σ, f 7→ fσ,where fσ is the σ-mean value function of f . These two spaces are smoothPΩ-representations, and the above surjective homomorphism is a homomor-phism of NF-spaces and PΩ-representations.The L(S(Y, V ), V ), L(SIndYP σ, V ) are the Schwartz distribution spaces,and they have the diagonal PΩ-actions on them which make them intosmooth PΩ-representations. The V -transpose of the above surjective homo-morphism is an injective homomorphism of TVS and PΩ-representations:L(SIndYP σ, V ) ↪→ L(S(Y, V ), V ).139Restriction of Distributions to Open SubsetsLet Y = GΩ≥w, GΩ>w or G, they are all open in G. We have the twocommutative diagrams of TVS and PΩ-representations in Figure 5.1. (Re-member that restriction maps of the pseudo-sheaf L(S(−, V ), V ) are denotedby res, while restriction maps of the pseudo-sheaf L(SInd−Pσ, V ) are denotedby Res.)On the left diagram (a), the four inclusion maps (vertical arrows) areinduced by the inclusions GΩ>w ⊂ GΩ≥w ⊂ G. The three horizontal maps aresurjective homomorphisms of TVS by the definition of Schwartz inductions.All six spaces in the left diagram (a) have the right regular PΩ-actions,and all maps in the left diagram (a) are PΩ-equivariant, hence they arePΩ-intertwining operators between smooth PΩ-representations.The right diagram (b) is the V -transpose of the left diagram (a). All fourvertical arrows are restrictions of distributions, and three horizontal arrowsare injective homomorphisms of TVS. All six distribution spaces have thediagonal PΩ-actions, and all maps in diagram (b) are intertwining operatorsof PΩ-representations.IΩ>w S(GΩ>w, V )IΩ≥w S(GΩ≥w, V )I S(G,V )(a) Schwartz spacesL(IΩ>w, V ) L(S(GΩ>w, V ), V )L(IΩ≥w, V ) L(S(GΩ≥w, V ), V )L(I, V ) L(S(G,V ), V )ResGGΩ≥wresGGΩ≥wResGΩ≥wGΩ>wresGΩ≥wGΩ>w(b) Distribution spacesFigure 5.1: Schwartz and Distribution-OpenExtension of Distributions from Closed SubsetsLet Y = GΩw, GΩ≥w or GΩ>w. The GΩ≥w, GΩ>w are open in G, while GΩwis closed in GΩ≥w. We have the two commutative diagrams of TVS andPΩ-representations in Figure 5.2.140On the left diagram (a), the above two vertical arrows are inclusionsof TVS induced by the inclusion GΩ>w ⊂ GΩ≥w. The two vertical arrowsbelow are surjective homomorphisms of TVS, given by restriction map toclosed subvariety GΩw. The three horizontal arrows are surjective homomor-phisms of TVS by the definition of Schwartz inductions. The compositionof two consecutive vertical arrows are zero, i.e. each vertical sequence indiagram (a) is a complex. All six spaces in diagram (a) have the right reg-ular PΩ-actions, and all maps in the diagram (a) are PΩ-equivariant henceintertwining operators between smooth PΩ-representations.The right diagram (b) is the V -transpose of the diagram (a). In dia-gram (b): The two vertical arrows above are restriction maps of distribu-tions. The two bottom vertical arrows are “extension of distributions fromclosed subsets” and they are injective homomorphisms of TVS. The threehorizontal maps are inclusions of distributions spaces. All six distributionspaces in diagram (b) have the diagonal PΩ-actions and they are smoothPΩ-representations. All maps in diagram (b) are PΩ-intertwining operators.The compositions of two consecutive vertical arrows are zero.IΩ>w S(GΩ>w, V )IΩ≥w S(GΩ≥w, V )IΩw S(GΩw, V )(a) Schwartz spacesL(IΩ>w, V ) L(S(GΩ>w, V ), V )L(IΩ≥w, V ) L(S(GΩ≥w, V ), V )L(IΩw , V ) L(S(GΩw, V ), V )ResGΩ≥wGΩ>wresGΩ≥wGΩ>w(b) Distribution spacesFigure 5.2: Schwartz and Distribution-Closed5.5 The Irreducibility of Unitary ParabolicInductionsIn this section, we keep the notations as in the last section, and let (σ, V )be a Harish-Chandra representation, and I = C∞IndGPσ = SIndGPσ be the141Schwartz induction (also the smooth induction). We want to study whenthe following equality holds:HomG(I, I) = C,i.e. the space of self-intertwining distributions is one dimensional.Remark 5.22. When the σ is of the form σ = τ⊗δ1/2P for a unitary (Hilbert)representation (τ, V ), the smooth induction I = C∞IndGPσ is infinitesimallyequivalent to the normalized unitary induction IndGP τ . If two representa-tions are infinitesimally equivalent, then they are irreducible/reducible si-multaneously. Moreover, we have HomG(I, I) = HomG(IndGP τ, IndGP τ) andIndGP τ is irreducible if and only if HomG(IndGP τ, IndGP τ) = C. Therefore, ifσ = τ ⊗ δ1/2P for a unitary representation τ , the following statements areequivalent:• The I = C∞IndGPσ = SIndGPσ is irreducible;• The IndGP τ is irreducible;• The HomG(IndGP τ, IndGP τ) = C;• HomG(I, I) = C.When the σ = τ ⊗ δ1/2P , to show the I is irreducible, it is sufficient to showHomG(I, I) = C.In this section, we find sufficient conditions for HomG(I, I) to be 1-dimensional, and in the Chapter 9, we will apply the results in this sectionto study the irreducibility of unitary inductions.Let D ∈ HomP (I, V ) be an intertwining distribution. For each fixedsubset Ω ⊂ Θ, we have seen its support suppD is a Zariski closed union of(P, PΩ)-double cosets. Then one has two possibilities of its support:• the extreme case: the support is contained in the identity doublecoset P ;• the general case: the support is not contained in the identity doublecoset P .We will discuss these two cases separately in following subsections.1425.5.1 General Case suppD * PSuppose the support suppD is not contained in P , then for each fixedsubset Ω ⊂ Θ, there exists a maximal (P, PΩ)-double coset contained insuppD, say GΩw for some w ∈ [WΘ\W/WΩ], w 6= e.For this particular Ω and w, let IΩ≥w, IΩ>w be the two local Schwartzinductions on GΩ≥w and GΩ>w respectively, and one has the inclusions IΩ>w ⊂IΩ≥w ⊂ I. By Lemma 5.14, we see the restriction of D to the subspace IΩ>wis zero, and the restriction of D to the subspace IΩ≥w is nonzero. We denotethe restriction of D to IΩ≥w by DΩ≥w, then it is in the up-left space in thefollowing diagramHomPΩ(IΩ≥w, V ) HomPΩ(S(GΩ≥w, V ), V )HomPΩ(IΩ>w, V ) HomPΩ(S(GΩ>w, V ), V )restriction restrictionMoreover the DΩ≥w ∈ HomPΩ(IΩ≥w, V ) is a nonzero element in the kernel ofthe restriction mapResΩw : HomPΩ(IΩ≥w, V )→ HomPΩ(IΩ>w, V )Since DΩ≥w vanishes on the subspace IΩ>w of IΩ≥w, it factor through aPΩ-equivariant continuous linear map on the quotient:DΩ≥w : IΩ≥w/IΩ>w → Vwhich makes the following diagram commuteIΩ≥w VIΩ≥w/IΩ>wDΩ≥wquotientDΩ≥wTo summarize the above, we have the following lemma which gives alocal description of intertwining distributions which have GΩw as a maximal(P, PΩ)-double coset in their supports:Lemma 5.23. Let D ∈ HomP (I, V ) be an intertwining distribution withsupport suppD.143(1) If GΩw is a maximal double coset in suppD, then the element DΩ≥w definedabove is a nonzero element in Ker(ResΩw).(2) The correspondenceKer(ResΩw)→ HomPΩ(IΩ≥w/IΩ>w, V )is a linear isomorphism. Moreover, since V, IΩ≥w, IΩ>w are nuclear hencereflexive, we haveHomPΩ(IΩ≥w/IΩ>w, V ) ' HomPΩ(V ′, (IΩ≥w/IΩ>w)′)where V ′, (IΩ≥w/IΩ>w)′ are the dual representation of V and IΩ≥w/IΩ>wrespectively.5.5.2 Extreme Case suppD ⊂ PIn this case, it does not matter what Ω is, since for all Ω, one hasGΩe = P,GΩ≥e = G,GΩ>e = G− P , andIΩ≥e = I≥e = IIΩ>e = I>e = SIndG−PP σThus we simply drop the superscript Ω.Since suppD ⊂ P , we have vanD ⊃ G − P , equivalently the D ∈HomP (I, V ) vanishes on the subspace I>e (which is also a subrepresentationof I). Therefore the D factor through the quotient representation I/I>e,and gives a P -equivariant mapD : I/I>e → Vwhich makes the following diagram commuteI VI/I>eDquotientDSimilar to the general case, we have the following local description ofintertwining distributions supported in P :144Lemma 5.24. Let D ∈ HomP (I, V ) be an intertwining distribution withsupport suppD. The correspondence{D ∈ HomP (I, V ) : suppD ⊂ P} → HomP (I/I>e, V )D 7→ Dis a one-to-one linear map. Moreover, since the V, I, I>e are all nuclearhence reflexive, we haveHomP (I/I>e, V ) ' HomP (V ′, (I/I>e)′),where the V ′ and (I/I>e)′ means the contragredient representation of V andI/I>e respectively.In sum, the intertwining distributions with supports contained in P , arein one-to-one correspondence with the space HomP (V′, (I/I>e)′).5.5.3 Irreducibility of IWe summarize the above two lemma about local descriptions of inter-twining distributions:Theorem 5.25. Let D ∈ HomG(I, I) = HomP (I, V ) be an intertwiningdistribution. Then either suppD ⊂ P or suppD * P .(1) If suppD ⊂ P , then the adjoint map of D is in the spaceHomP (V′, (I/I>e)′).(2) If suppD * P , then for each Ω ⊂ Θ, there exists a w ∈ [WΘ\W/WΩ],where w 6= e, such that the adjoint map of (restricted distribution) DΩ≥wis a nonzero element inHomPΩ(V′, (IΩ≥w/IΩ>w)′)Remark 5.26. Combining with Lemma 5.21, to show HomG(I, I) = C,(equivalently I is irreducible when σ = τ ⊗ δ1/2P for a unitary τ), we justneed to:(1) ShowHomP (V′, (I/I>e)′) = C.145(2) Find a subset Ω ⊂ Θ, and show for all w ∈ [WΘ\W/WΩ], w 6= e, onehasHomPΩ(V′, (IΩ≥w/IΩ>w)′) = {0}.(Therefore it contains no nonzero elements.)The (2) guarantees that there is no intertwining distribution with sup-ports not contained in P . Otherwise supposeD has its support not containedin P , then for every Ω, one can find a maximal double coset GΩw in suppD,and a nonzero element in some HomPΩ(V′, (IΩ≥w/IΩ>w)′), a contradiction!Then the (1) guarantees that the only possible intertwining distributionsare scalar intertwining distributions, therefore HomG(I, I) = HomP (I, V ) =C.146Chapter 6Schwartz DistributionsSupported in Double CosetsSummary of This ChapterIn the last chapter, we are required to study the spaces(IΩ≥w/IΩ>w)′for various Ω ⊂ Θ and w ∈ [WΘ\W/WΩ]. Recall that we have the followingexact sequence of NF-spaces0→ IΩ>w → IΩ≥w → IΩ≥w/IΩ>w → 0Since all spaces are NF-spaces, its dual sequence is still exact:0→ (IΩ≥w/IΩ>w)′ → (IΩ≥w)′ → (IΩ>w)′ → 0The space (IΩ≥w/IΩ>w)′ is exactly the kernel of the restriction map (IΩ≥w)′ →(IΩ>w)′. We will show the kernel of the restriction map (IΩ≥w)′ → (IΩ>w)′consists of transverse derivatives of distributions on GΩw. In this chapter, wewill study the case Ω = ∅ and the general cases of Ω will be studied in thelast chapter.Notations and SettingsLet G,P = PΘ, P∅,W,WΘ, [WΘ\W ], (σ, V ) be the same as in Chapter5. The (P, P∅)-double cosets in G are parameterized by the set [WΘ\W ]of minimal representatives, and they are ordered by the closure order (seeDefinition 3.28).For a w ∈ [WΘ\W ], let Gw, G≥w, G>w and Iw, I≥w, I>w be the sameas in 5.4. Remember that G≥w, G>w are Zariski open in G and Gw isthe complement of G>w in G≥w hence Gw is closed in G≥w. The localSchwartz induction I>w is a subspace of I≥w, and the restriction to Gwgives a surjective homomorphism I≥w → Iw.147We will work on a more general setting. LetF = a nuclear Fre´chet spaceand letS(−, V ) = the pseudo-cosheaf of V -valued Schwartz functions on GL(S(−, V ), F ) = the pseudo-sheaf of F -valued Schwartz distributions on GWe use the following notations to denote the extension maps (by zero)of Schwartz V -valued functions and Schwartz inductions:exG≥wG>w: S(G>w, V ) ↪→ S(G≥w, V )ExG≥wG>w: I>w ↪→ I≥wand the following notations to denote the restriction maps of Schwartz F -valued distributions:resG≥wG>w: L(S(G≥w, V ), F )→ L(S(G>w, V ), F )ResG≥wG>w: L(I≥w, F )→ L(I>w, F )and for the special case F = C:resw : S(G≥w, V )′ → S(G>w, V )′Resw : I′≥w → I ′>wThey fit into the following diagramKer(resG≥wG>w) L(S(G≥w, V ), F ) L(S(G>w, V ), F )Ker(ResG≥wG>w) L(I≥w, F ) L(I>w, F )resG≥wG>wResG≥wG>wFigure 6.1: Restriction maps and kernelsThe two leftmost kernels (for the case F = C) are the spaces we need tostudy in this chapter.148New Notations in This ChapterLet NP be the opposite unipotent radical of P . We introduce the fol-lowing new notations:Nw = w˙−1NP w˙N+w = w˙−1NP w˙ ∩N∅= Nw ∩N∅N−w = w˙−1NP w˙ ∩N∅= Nw ∩N∅The N+w , N−w are closed subgroups of Nw and N+w ∩N−w = {e}, and the mul-tiplication map N+w ×N−w → Nw is a smooth diffeomorphism and an isomor-phism of real algebraic varieties (not a group isomorphism). Let nw, n+w , n−wbe their complexified Lie algebras respectively, and one has nw = n+w + n−w .Main Theorem of This ChapterThe main theorem in this chapter is (see 6.1.1 for more details):Theorem (Theorem 6.1). The right multiplication of U(n−w) (as deriva-tives), on the distribution spaces Ker(resG≥wG>w) and Ker(ResG≥wG>w), gives thefollowing isomorphisms (of TVS):L(S(Gw, V ), F )⊗ U(n−w) ∼−→ Ker(resG≥wG>w)L(Iw, F )⊗ U(n−w) ∼−→ Ker(ResG≥wG>w)In this thesis, we will only need the case F = C, but we prefer to includethe general result since there is no essential difficulty to prove the generalcase.This theorem means, Schwartz distributions supported in Gw, are ex-actly the transverse derivatives of Schwartz distributions on Gw. We willexplain the term “transverse” in 7.1.5 of the next chapter.6.1 Preparation• In 6.1.1, we first formulate the main theorem in this chapter, namelyexplain the meaning of the maps (6.1) and (6.2) in the main theorem.• In 6.1.2, we construct a Zariski open tubular neighbourhood Zw of theGw.149• In 6.1.3, we change the neighbourhood from G≥w to Zw. Since theSchwartz distributions have the pseudo-sheaf property, the space ofSchwartz distributions supported in Gw is independent of the choiceof neighbourhoods (Lemma 4.38 and Lemma 4.80).6.1.1 Formulating the Main TheoremWe explain the main theorem at the beginning of this chapter. Remem-ber we want to study the algebraic structure of the following two kernels ofrestriction maps of distributions:Ker(resG≥wG>w) = Ker{L(S(G≥w, V ), F )→ L(S(G>w, V ), F )}Ker(ResG≥wG>w) = Ker{L(I≥w, F )→ L(I>w, F )}where F is an arbitrary NF-space. (Here we abuse the notation, and usethe same Res, res to denote restrictions with different target spaces F . Thiswill not create ambiguity since we will only work on a single target space Fat each time.)Finally we will see for different F , they all have similar structure and weonly need to study the case F = C.The Kernel Ker{L(S(G≥w, V ), F )→ L(S(G>w, V ), F )}The G≥w and G>w are Zariski open subset of G, hence the U(g) acts onthem as algebraic differential operators, and makes the Schwartz functionspaces S(G≥w, V ) and S(G>w, V ) into left U(g)-modules. The inclusionmapS(G>w, V ) ↪→ S(G≥w, V )is a left U(g)-homomorphism.The transpose action of differential operators makes the distributionspaces L(S(G≥w, V ), F ) and L(S(G>w, V ), F ) into right U(g)-modules. Andthe restriction mapresG≥wG>w: L(S(G≥w, V ), F )→ L(S(G>w, V ), F )is a right U(g)-homomorphism. Hence the kernelKer(resG≥wG>w) = Ker{L(S(G≥w, V ), F )→ L(S(G>w, V ), F )}is also a right U(g)-module (submodule of L(S(G≥w, V ), F ).)150Since Gw is closed in G≥w, by Lemma 4.36, we have the following inclu-sion of TVS:L(S(Gw, V ), F ) ↪→ L(S(G≥w, V ), F ).We regard the L(S(Gw, V ), F ) as a subspace of L(S(G≥w, V ), F ). Thissubspace is contained in the kernel Ker(resG≥wG>w), since the compositionS(G>w, V ) ↪→ S(G≥w, V ) S(Gw, V )is zero.The U(n−w) is a subalgebra of U(g), and the right multiplication of (al-gebraic differential operators) on distribution space gives the following mapL(S(Gw, V ), F )⊗ U(n−w)→ Ker(resG≥wG>w) (6.1)Φ⊗ u 7→ Φ · uwhere the distribution Φ · u is given by〈Φ · u, f〉 := 〈Φ, (Ru · f)|Gw〉.The Kernel Ker{L(I≥w, F )→ L(I>w, F )}The situation for distributions on Schwartz inductions is similar. TheG≥w, G>w are in the Zariski P -topology of G, hence we have the inclusionof left U(g)-modulesI>w ↪→ I≥w.And its F -transpose gives the restriction mapResG≥wG>w: L(I≥w, F )→ L(I>w, F )which is a right U(g) homomorphism, with its kernel Ker(ResG≥wG>w) a rightU(g)-module (submodule).Since Gw is closed in G≥w and both are left P -stable, by Lemma 4.82,we have the injective homomorphismL(Iw, F ) ↪→ L(I≥w, F )with its image contained in the kernel Ker(ResG≥wG>w), since the compositionI>w ↪→ I≥w Iw is zero.151The right multiplication of the subalgebra U(n−w) of U(g) on the subspaceL(Iw, F ) gives the following mapL(Iw, F )⊗ U(n−w)→ L(I≥w, F ) (6.2)Φ⊗ u 7→ Φ · uwhere the Φ · u is given by〈Φ · u, φ〉 := 〈Φ, (Ruφ)|Gw〉, ∀φ ∈ I≥w = SIndG≥wP σ.The Main TheoremThe main theorem of this chapter is;Theorem 6.1. The map (6.1) and (6.2) are isomorphisms of TVS.Remark 6.2. The U(n−w) is endowed with the inductive limit topology ofUn(n−w) and is a LF-space and nuclear space. Here Un(n−w) means the finitedimensional subspace spanned by n products of elements in n−w .6.1.2 Gw and its Tubular Neighbourhood ZwThe G≥w is an open neighbourhood of Gw, but its structure is hard todescribe. We introduce another open neighbourhood Zw of Gw which is atubular neighbourhood.The Neighbourhood ZwFor each w ∈ [WΘ\W ], remember we abuse the notation and denote thefixed representative of it in G by the same w. Let Nw = w−1NPw be as thebeginning of this chapter. We letZw = PwNw = PNPw,i.e. it is the (P,Nw)-double coset through the base point w, and it is exactlythe right translation of PNP by w.Lemma 6.3. The Zw is a Zariski open subset of G. And the mapP ×Nw → Zw (6.3)(p, n) 7→ pwnis an isomorphism of varieties and manifolds.152The Structure of GwRecall that Gw = PwP∅ is the (P, P∅)-double coset through the basepoint w. It is easy to see Gw is also the (P,N∅)-double coset through w:Gw = PwN∅,since wM∅w−1 = M∅ for all w ∈W .The N∅ is the product of two subgroups, i.e. the multiplication map(N∅ ∩ w−1Pw)× (N∅ ∩ w−1NPw)→ N∅ (6.4)(n1, n2) 7→ n1n2is an isomorphism of real algebraic variety and smooth manifolds (NOT agroup isomorphism). Since the w-conjugation of first part is contained inP , we seeGw = Pw(N∅ ∩ w−1NPw) = PwN+wwhere N+w = N∅ ∩ w−1NPw as the beginning of this chapter.Lemma 6.4. The mapP ×N+w → Gw (6.5)(p, n) 7→ (pwn)is an isomorphism of varieties and manifolds. Hence the Gw is a nonsingularclosed subvariety of Zw and closed regular submanifold of Zw. The followingdiagram commutes in the category of real algebraic varieties and smoothmanifoldsP ×Nw ZwP ×N+w Gw(6.3)(6.5)Here the two vertical arrows are closed embeddings and the two horizontalarrows are isomorphisms.6.1.3 Change of NeighbourhoodsWe now have two Zariski P -open neighbourhoods of Gw: G≥w and Zw,hence the intersection G≥w ∩ Zw is also a Zariski P -open neighbourhood of153Gw. The complements of Gw in them are G>w, Zw−Gw and G≥w∩Zw−Gwrespectively (all three complements are Zariski P -open in G).We then have the following two commutative diagrams, where all hori-zontal arrows are restriction maps of distributions, and all vertical sequencesare exact:0 0 0Ker(resZwZw−Gw) Ker(resZw∩G≥wZw∩G≥w−Gw) Ker(resG≥wG>w)L(S(Zw, V ), F ) L(S(Zw ∩G≥w, V ), F ) L(S(G≥w, V ), F )L(S(Zw −Gw, V ), F ) L(S(Zw ∩G≥w −Gw, V ), F ) L(S(G>w, V ), F )0 0 0(a) (b)resresZwZw−Gw resZw∩G≥wZw∩G≥w−GwresresG≥wG>wres res0 0 0Ker(ResZwZw−Gw) Ker(ResZw∩G≥wZw∩G≥w−Gw) Ker(ResG≥wG>w)L(SIndZwP σ, F ) L(SIndZw∩G≥wP σ, F ) L(I≥w, F )L(SIndZw−GwP σ, F ) L(SIndZw∩G≥w−GwP σ, F ) L(I>w, F )0 0 0(c) (d)ResResZwZw−Gw ResZw∩G≥wZw∩G≥w−GwResResG≥wG>wRes ResBy Lemma 4.38, the maps (a)(b) are isomorphisms of TVS, and byLemma 4.80, the maps (c)(d) are isomorphisms of TVS. Hence as in Lemma4.39 and Lemma 4.81, we have the following isomorphisms of kernels of154restriction maps:Ker(resG≥wG>w) ' Ker(resZwZw−Gw) (6.6)Ker(ResG≥wG>w) ' Ker(ResZwZw−Gw)Remark 6.5. As we have seen in the last subsection, the Zw is a tubularneighbourhood of Gw and we have good product decomposition of both Gwand Zw in Lemma 6.3 and Lemma 6.4. This makes the analysis of thekernels Ker(resZwZw−Gw) and Ker(ResZwZw−Gw) much easier than the originalkernels Ker(resG≥wG>w) and Ker(ResG≥wG>w).6.2 Schwartz Distributions with Point SupportsIn this section, letN = a unipotent algebraic group defined over RN = N(R)e = the identity of NN∗ = N − {e}n0 = the real Lie algebra of Nn = the complexification of n0The N is a nonsingular affine real algebraic variety and affine Nash manifold,we have the pseudo-cosheaf S(−,C) of C-valued Schwartz functions on N ,and the pseudo-sheaf S(−,C)′ of Schwartz distributions on N . The {e} isa nonsingular closed subvariety of N and a closed Nash submanifold of N ,the N∗ is Zariski open and restricted open in N .Since N∗ is Zariski open in N , we have the inclusion mapS(N∗,C) ↪→ S(N,C)under which the S(N∗,C) is identified with the subspace{f ∈ S(N,C) : Df(e) = 0,∀D ∈ D(N)}of Schwartz functions vanishing with all derivatives at e.Remark 6.6. Here the D is the sheaf of (complexified) algebraic differentialoperators, and the D(N) is the (complexified) ring of algebraic differentialoperators on N . Since N is affine real algebraic variety, the D(N) generates155the ring of (complexified) Nash differential operators over the ring of complexvalued Nash functions. Hence a Schwartz function vanishes with all Nashderivatives at e, is equivalent to say it vanishes with all algebraic derivativesat e. In this section, all geometric spaces are affine real algebraic varieties,hence it is safe to work only on algebraic derivatives.Also if not otherwise stated, all differential operators are complexified,since we work on complex valued smooth (Schwartz) functions.The transpose of the above inclusion map is the restriction map of dis-tributions, denoted byres : S(N,C)′ → S(N∗,C)′.In this section, we study the kernel of this restriction map. It is a closedsubspace of S(N,C)′, and fits into the exact sequence0→ Ker(res)→ S(N,C)′ res−−→ S(N∗,C)′ → 0 (6.7)The main result in this section isLemma 6.7. The kernel Ker(res) is isomorphic to U(n). More precisely,the following map is an isomorphism of TVS:U(n) 7→ Ker(res) (6.8)u 7→ Ωe · uwhere Ωe · u is given by〈Ωe · u, f〉 := 〈Ωe, Ruf〉 = (Ruf)(e)for all f ∈ S(N,C). And the U(n) is endowed with the direct limit topologyfrom finite dimensional subspaces Un(n).We omit the proof since it is exactly the same as the proof of TheoremXXXVI on page 101 of [35].6.3 Proof of the Main TheoremWe first prove the theorem for the case F = C and we use the followingsimplified notations:resw : S(Zw, V )′ → S(Zw −Gw, V )′Resw : (SIndZwP σ)′ → (SIndZw−GwP σ)′156i.e. they are the restriction maps of C-valued distributions on Schwartzfunctions and Schwartz inductions respectively. In this section, we show thefollowing special case (F = C) of the main theorem:Lemma 6.8. The following maps, given by multiplication of derivatives inU(n−w), are isomorphisms of TVS:S(Gw, V )′ ⊗ U(n−w)→ Ker(resw), Φ⊗ u 7→ Φ · u (6.9)(SIndGwP σ)′ ⊗ U(n−w)→ Ker(Resw), Φ⊗ u 7→ Φ · u (6.10)More precisely, let Φ ∈ S(Gw, V )′ (resp. (SIndGwP σ)′) and u ∈ U(n−w), theΦ · u is〈Φ · u, f〉 = 〈Φ, (Ruf)|Gw〉for all f ∈ S(Zw, V ) (resp. SIndZwP σ).Later we will use this Lemma to show the main theorem for generalNF-spaces F .6.3.1 The Kernel Ker(resw)Letresw : S(Zw, V )′ → S(Zw −Gw, V )′be the restriction map of C-valued Schwartz V -distributions on Zw, as inthe beginning of this section.Isomorphisms between Varieties (Affine Nash Manifolds)Combining the Lemma 6.3 and Lemma 6.4 and the isomorphism N+w ×N−w∼−→ Nw, we have the following isomorphisms:P ×N+w ×N−w ∼−→ ZwP ×N+w × (N−w − {e}) ∼−→ Zw −GwP ×N+w × {e} ∼−→ GwThen by (E-6) of Proposition 4.30, we have the following isomorphisms ofSchwartz V -valued function spaces:S(P ×N+w , V ) ⊗̂ S(N−w ,C) ∼−→ S(Zw, V )S(P ×N+w , V ) ⊗̂ S(N−w − {e},C) ∼−→ S(Zw −Gw, V )S(P ×N+w , V ) ⊗̂ S({e},C) ∼−→ S(Gw, V )157Isomorphisms between Distribution SpacesSince all the above Schwartz spaces are nuclear, we have the correspond-ing isomorphisms on their strong dual:S(Zw, V )′ ∼−→ S(P ×N+w , V )′ ⊗̂ S(N−w ,C)′S(Zw −Gw, V )′ ∼−→ S(P ×N+w , V )′ ⊗̂ S(N−w − {e},C)′S(Gw, V )′ ∼−→ S(P ×N+w , V )′ ⊗̂ S({e},C)′Note that S({e},C) = C, i.e. Schwartz functions at a point are exactly“constant”, and S({e},C)′ = CΩ{e}e is the one dimensional space spannedby the “delta function” on {e}.It is easy to verify that the following diagrams commutes:S(Zw, V )′ S(P ×N+w , V )′ ⊗̂ S(N−w ,C)′S(Zw −Gw, V )′ S(P ×N+w , V )′ ⊗̂ S(N−w − {e},C)′'resw id⊗rw'Here the rightmost vertical arrow is the tensor product of the identity mapon S(P ×N+w , V )′ with the restriction maprw : S(N−w ,C)′ → S(N−w − {e},C)′.Therefore, we have the isomorphism between the kernels of the abovetwo vertical maps:Ker(resw) ' S(P ×N+w , V )′ ⊗̂Ker(rw).The Tensor Product S(P ×N+w , V )′ ⊗Ker(rw)In section 6.2, we have seen the following isomorphism between TVSU(n−w)∼−→ Ker(rw),where U(n−w) is given the limit topology from the finite dimensional sub-spaces Un(n−w), and is a LF-space.158By Lemma 2.13, we haveKer(resw) ' S(P ×N+w , V )′ ⊗̂ U(n−w)= S(P ×N+w , V )′ ⊗̂ lim−→nUn(n−w)= lim−→n[S(P ×N+w , V )′ ⊗̂ Un(n−w)]= lim−→n[S(P ×N+w , V )′ ⊗ Un(n−w)]= S(P ×N+w , V )′ ⊗ U(n−w)In other word, the algebraic tensor product S(P×N+w , V )′⊗U(n−w) is alreadycomplete, and we don’t need the completion on the ⊗.In sum, we have an isomorphism of TVSKer(resw) ' S(P ×N+w , V )′ ⊗ U(n−w) (6.11)6.3.2 The Isomorphism (6.9)In the last subsection, we have shown the isomorphism (6.11):Ker(resw) ' S(P ×N+w , V )′ ⊗ U(n−w)which is further isomorphic to S(Gw, V )′⊗U(n−w) since S(Gw, V )′ ' S(P ×N+w , V )′. Hence To show the map (6.9) is an isomorphism, we just need toverify it is the same as the isomorphism (6.11).More precisely, we just need to verify that the following diagram com-mutesS(P ×N+w , V )′ ⊗ U(n−w)S(P ×N+w , V )′ ⊗̂Ker(rw) Ker(resw)S(P ×N+w , V )′ ⊗ S(N−w ,C)′ S(Zw, V )′(S(P ×N+w , V )⊗ S(N−w ,C))′This is routine and we omit the detail. One just need to pick a functionof the form φ ⊗ ψ for φ ∈ S(P × N+w , V ) and ψ ∈ S(N−w ,C), and check159the value 〈Φ · u, φ ⊗ ψ〉 through two ways, and see they are equal. Thecrucial reason is the Lie algebra n−w acts on Schwartz functions through leftinvariant vector fields, i.e. by differentiation of the right regular Nw-action,and it will not affect the P -component on the left.6.3.3 The Kernel Ker(Resw) and the Isomorphism (6.10)To show the map (6.10) is an isomorphism, we just need to combinethe isomorphism U(n−w) ' Ker{S(N−w ,C) → S(N−w − {e},C)} with theisomorphism in Lemma 4.91.By Lemma 4.91, we have the following isomorphismsSIndZwP σ ' SIndPPσ ⊗̂ S(Nw,C)SIndZw−GwP σ ' SIndPPσ ⊗̂ S(Nw −N+w ,C)and the isomorphisms on their dual(SIndZwP σ)′ ' (SIndPPσ)′ ⊗̂ S(Nw,C)′(SIndZw−GwP σ)′ ' (SIndPPσ)′ ⊗̂ S(Nw −N+w ,C)′Thus the following diagram commutes:(SIndZwP σ)′ (SIndZw−GwP σ)′(SIndPPσ)′ ⊗̂ S(Nw,C)′ (SIndPPσ)′ ⊗̂ S(Nw −N+w ,C)′Resw' 'where the bottom arrow is the tensor product map id ⊗ resNwNw−N+w . Hencewe have the isomorphism on the kernelsKer(Resw) ' (SIndPPσ)′ ⊗̂Ker(resNwNw−N+w ).ThereforeKer(Resw) ' (SIndPPσ)′ ⊗̂Ker(resNwNw−N+w )' (SIndPPσ)′ ⊗̂ S(N+w ,C)′⊗̂Ker{S(N−w ,C)′ → S(N−w − {e},C)′}' (SIndPPσ)′ ⊗̂ S(N+w ,C)′ ⊗̂ U(n−w)' (SIndGwP σ)′ ⊗̂ U(n−w)And by the same argument as in the last subsection, we can verify thisisomorphism is exactly the map (6.10), hence (6.10) is an isomorphism ofTVS. The proof of Lemma 6.8 is completed.1606.3.4 General Case of FWe prove the general case for arbitrary nuclear Fre´chet space F , althoughwe won’t use this result in the thesis. We only show the isomorphismL(Iw, F )⊗ U(n−w) ' Ker{ResG≥wG>w : L(I≥w, F )→ L(I>w, F )}since the other isomorphism is similar.First we have the isomorphism:Ker{ResG≥wG>w : L(I≥w, F )→ L(I>w, F )} ' Ker{I ′≥w → I ′>w} ⊗̂ FSecond we have the natural maps(I ′w ⊗ U(n−w))⊗ F ∼−→ (I ′w ⊗ F )⊗ U(n−w)↪→ (I ′w ⊗̂ F )⊗ U(n−w)∼−→ L(Iw, F )⊗ U(n−w)and their composition extends to a continuous linear map(I ′w ⊗ U(n−w)) ⊗̂ F → L(Iw, F )⊗ U(n−w),which makes the following diagram commute(I ′w ⊗ U(n−w)) ⊗̂ F Ker{I ′≥w → I ′>w} ⊗̂ FL(Iw, F )⊗ U(n−w) Ker{L(I≥w, F )→ L(I>w, F )}∼∼To show the bottom arrow is an isomorphism, we just need to show theleft vertical arrow is an isomorphism. Actually(I ′w ⊗ U(n−w)) ⊗̂ F = lim−→n[I ′w ⊗ Un(n−w)] ⊗̂ F' lim−→n[(I ′w ⊗ Un(n−w)) ⊗̂ F ]' lim−→n[(I ′w ⊗̂ F )⊗ Un(n−w)]= lim−→n[L(Iw, F )⊗ Un(n−w)]' L(Iw, F )⊗ U(n−w)(Here we use Lemma 2.13 twice in the first and last steps.)161Chapter 7Torsions of DistributionsSummary of This ChapterIn the last chapter, we have shown the spaces (I≥w/I>w)′ are isomorphictoI ′w ⊗ U(n−w)for all w ∈ [WΘ\W ]. In this chapter, we continue to study the n∅-torsionsubspaces of them, and show their n∅-torsion subspaces are given by thetorsion subspace on the first component I ′w.Notations in This ChapterAs in Chapter 6, we use the following notations to denote the restrictionmaps of (scalar valued) distributions:resw : S(G≥w, V )′ → S(G>w, V )′Resw : I′≥w → I ′>wThese two restriction maps are P∅-homomorphisms of P∅-representations,when the four dual spaces are endowed with the contragredient actions ofP∅. We have seen thatKer(resw) = (S(G≥w, V )/S(G>w, V ))′Ker(Resw) = (I≥w/I>w)′and these two spaces are the main objects studied in the last chapter.For each n ∈ Z≥0, let Un(n−w) be the (finite dimensional) subspace ofU(n−w) spanned by products of k-elements in n−w for all k ≤ n. By convention,let U0(n−w) = C. The {Un(n−w), n ≥ 0} is an exhaustive filtration on U(n−w).The Main Theorem of This ChapterThe main theorem of this chapter is (see 7.1.1 for details):162Theorem (Theorem 7.1). The n∅-torsion subspaces of the two kernels aregiven by:(Ker(resw))[n∅•] = [S(Gw, V )′ ⊗ U(n−w)][n∅•] = [S(Gw, V )′][n∅•] ⊗ U(n−w)(Ker(Resw))[n∅•] = [I ′w ⊗ U(n−w)][n∅•] = [I ′w][n∅•] ⊗ U(n−w)Moreover, these two equalities are M∅-equivariant when both sides are en-dowed with the obvious M∅-actions.Reading Guide• Section 7.1 consists of preliminary works. We explain the main theo-rem with more details, show a product formula in enveloping algebrawhich will be frequently used, and discuss the transverse decomposi-tion of tangent vector fields on the submanifolds Gw. In particular, weshow the first equality in the above main theorem implies the secondequality, and we just need to show the first one, namely[S(Gw, V )′ ⊗ U(n−w)][n∅•] = [S(Gw, V )′][n∅•] ⊗ U(n−w).• In 7.2, we show the inclusion[S(Gw, V )′ ⊗ U(n−w)][n∅•] ⊃ [S(Gw, V )′][n∅•] ⊗ U(n−w).This is the easy part of the main theorem, and is proved by purealgebraic tricks.• In 7.3, we show the reversed inclusion[S(Gw, V )′ ⊗ U(n−w)][n∅•] ⊂ [S(Gw, V )′][n∅•] ⊗ U(n−w).This is the most subtle part of the entire thesis. In a word, we choose a“good” linear order on the PBW-basis of the enveloping algebra U(n−w),filter the left-hand-side by this order, and prove each filtered part isincluded in the right-hand-side by induction on this linear order.7.1 PreparationThis section is the preparation work for the proof of the main theorem(Theorem 7.1). Each subsection could be read independently.163• In 7.1.1, we discuss the main theorem in more details. The entirechapter is reduced to proving the first equality in the main theorem:(S(Gw, V )′ ⊗ U(n−w))[n∅•] = [S(Gw, V )′][n∅•] ⊗ U(n−w).The second equality in the main theorem is implied by this equality.• In 7.1.2, we recall the notion of transpose modules over an envelopingalgebra. The U(g)-actions on the distribution spaces are essentiallythe transpose U(g)-module of the Schwartz function spaces. It is morenatural and convenient to consider the distribution spaces as rightU(g)-modules instead of left U(g)-modules. In the following sectionsof this chapter, we will regard all distribution spaces as right modules.• In 7.1.3, we introduce some simplified notations which are convenientfor our proof.• In 7.1.4, we show a formula in the enveloping algebra, which will beused repeatedly in the proof.• In 7.1.5, we recall the necessary geometric notions: left and right in-variant vector fields, transverse tangent fields, and show that the ele-ments in n−w are indeed transverse to the submanifold Gw.7.1.1 Main Theorem in this ChapterWe have seen the following commutative diagram0 S(G>w, V ) S(G≥w, V ) S(G≥w,V )S(G>w,V ) 00 I>w I≥wI≥wI>w0where the two rows are exact sequence of NF-spaces, the three verticallarrows are surjective and all arrows are P∅-equivariant. By taking the strongdual, we have the following commutative diagram of DNF-spaces (and dualP∅-representations), in which all two rows are exact sequences and threevertical arrows are inclusions:0 (S(G≥w,V )S(G>w,V ))′ S(G≥w, V )′ S(G>w, V )′ 00 (I≥wI>w)′ I ′≥w I′>w 0reswResw164The two leftmost spaces, namelyKer(resw) = (S(G≥w, V )/S(G>w, V ))′Ker(Resw) = (I≥w/I>w)′are the main object to study. The two kernels are P∅-subrepresentations ofS(G≥w, V )′ and I ′≥w respectively, and in particular they are also n∅-modules(actually even g-modules since G≥w is open), with the Lie algebras acting onthem as algebraic derivatives. In this chapter we will compute the n∅-torsionsubspaces on the two n∅-modules Ker(Resw) and Ker(resw).The Main Theorem in this ChapterIn Chapter 6, we have shown the Ker(resw) (resp. Ker(Resw)) is givenby U(n−w)-transverse derivatives of distributions in S(Gw, V )′ (resp. I ′w), i.e.we have the two horizontal linear isomorphisms in the following diagramS(Gw, V )′ ⊗ U(n−w) Ker(resw)I ′w ⊗ U(n−w) Ker(Resw)Figure 7.1: Ker(resw) and Ker(Resw) as transverse derivativesThe right side inclusion Ker(Resw) ↪→ Ker(resw) is P∅-equivariant (un-der the contragredient P∅-actions on S(G≥w, V )′ and I ′≥w respectively), inparticular it is a homomorphism of n∅-modules.The two horizontal isomorphisms transport the n∅-actions on the righttwo kernel spaces to the left two tensor products. The n∅ acts on themas derivatives, while the U(n−w) also acts as (transverse) derivatives, henceone regard U(n∅) and U(n−w) as subrings of the U(g), which also acts asderivatives on distribution spaces. In particular, one can multiply elementsin U(n∅) and U(n−w) inside U(g).These derivative actions are complicated, and there is no easy algebraicdescriptions. In other words, the tensor products S(Gw, V )′ ⊗ U(n−w) andI ′w⊗U(n−w) are not tensor products of n∅-modules, as one can see the U(n−w)is not stable under n∅-multiplication.However, their n∅-torsion subspaces, behave like tensor product modules:165Theorem 7.1. Under the two horizontal isomorphisms in Figure 7.1, then∅-torsion subspaces on Ker(resw) and Ker(Resw) are given by[Ker(resw)][n∅•] = (S(Gw, V )′ ⊗ U(n−w))[n∅•] = (S(Gw, V )′)[n∅•] ⊗ U(n−w)(7.1)[Ker(Resw)][n∅•] = (I ′w ⊗ U(n−w))[n∅•] = (I ′w)[n∅•] ⊗ U(n−w)Since the inclusion Ker(Resw) ↪→ Ker(resw) is an n∅-homomorphism, byLemma 3.36, we see[Ker(Resw)][n∅•] = Ker(Resw) ∩ [Ker(resw)][n∅•].If we can show the first equality in the above theorem, then we have[Ker(Resw)][n∅•] = Ker(Resw) ∩ [Ker(resw)][n∅•]= [I ′w ⊗ U(n−w)] ∩ [S(Gw, V )′ ⊗ U(n−w)][n∅•]= [I ′w ⊗ U(n−w)] ∩ {[S(Gw, V )′][n∅•] ⊗ U(n−w)}= {I ′w ∩ [S(Gw, V )′][n∅•]} ⊗ U(n−w)= [I ′w][n∅•] ⊗ U(n−w)Hence the second equality in the above theorem is implied by the first one,we just need to show the first equality. The rest of this chapter is devoted tothis object.Remark 7.2. Showing the second equality directly, is much more difficultthan imagined. The Schwartz induction spaces I>w, I≥w are included insmooth function spaces on corresponding algebraic manifolds, and one cantake derivatives of functions in them, however the resulting functions neednot satisfy the σ-rule. Although one can take the derivatives in Lie algebra,we will see in the following sections that derivatives in Lie algebra are notsufficient for the discussion.7.1.2 Left vs. RightOn Schwartz distribution spaces, the left action of Lie algebra is thetranspose action of the Lie algebra on the corresponding Schwartz functionspaces.For example, the S(G≥w, V ) is a left U(g)-module, since U(g) acts asalgebraic differential operators on G and G≥w is open in G. On the corre-sponding distribution space S(G≥w, V )′, the left U(g)-module structure is166given by the transpose U(g)-action:〈X · Φ, f〉 := 〈Φ,−X · f〉for all Φ ∈ S(G≥w, V )′, f ∈ S(G≥w, V ), X ∈ g. Here the −X · f =R−Xf is the differentiation of the right regular action. More generaly, forX1, . . . , Xk ∈ g, one has〈X1 · · ·Xk · Φ, f〉 = 〈Φ, (−1)kXk · · ·X1 · f〉.To get rid of the annoying (−1)k and the reversed order, it is morenatural to regard a distribution space as a right module over an envelopingalgebra.Remark 7.3. On distribution spaces, it is more natural to let differentialoperators act on the right, since the differential operators actually act onfunctions rather than distributions.On Schwartz function spaces, the right regular action is an action, andthe Lie algebra action makes them into left modules. There is no ambiguityabout left or right in discussion.Transpose Right Module of a Left ModuleDefinition 7.4 (Involution on Enveloping Algebra). Let h be an ar-bitrary complex Lie algebra, and U(h) be its enveloping algebra. The map−1 : h→ h, X 7→ −X induces an involution on the U(h)U(h)→ U(h)u 7→ tuMore precisely, letX1, . . . , Xk ∈ h be arbitrary elements, then t(X1 · · ·Xk) =(−1)k(Xk · · ·X1).Definition 7.5 (Transpose Module of Left U(h)-Module). Let M bea left U(h)-module. Then it has a natural structure of right U(h)-module,given bym · u := tu ·mfor all u ∈ U(h),m ∈M. We call M with this right U(h)-module structurethe transpose (right) U(h)-module of the left U(h)-module M.We keep the notation as in 3.4. Keep in mind that we use superscript fortorsion subspaces of left module, and subscript for torsions of right module.The right transpose U(h)-module of a left U(h)-module M share the sametorsion submodule with M, i.e. we have the following easy fact:167Lemma 7.6. Let M be a left U(h)-module, and by abuse of notation wedenote its transpose right U(h)-module by the same M. For k ∈ Z≥0, letM[hk] = the annihilator of (hk) in the left U(h)-module MM[hk] = the annihilator of (hk) in the transpose right U(h)-module MM[h•] = the h-torsion submodule of the left U(h)-module MM[h•] = the h-torsion submodule of the right transpose U(h)-module MThenM[hk] =M[hk] ∀k ≥ 0M[h•] =M[h•]Left Module vs. Right ModuleFor a Schwartz distribution space with appropriate left module struc-ture, e.g. the left U(p∅)-module S(Gw, V )′, or left U(g)-module S(G≥w, V )′and Ker(resw), we will consider their transpose right module, and torsionsubspaces.For example, on S(G≥w, V )′ or its submodule Ker(resw), the transposeright U(g)-action is〈Φ · u, f〉 = 〈tu · Φ, f〉= 〈Φ, t(tu) · f〉= 〈Φ, u · f〉for all Φ ∈ S(G≥w, V )′, f ∈ S(G≥w, V ), u ∈ U(g). In other words, the righttranspose action is exactly the right multiplication of derivatives.Similarly on S(Gw, V )′ we have the transpose right U(p∅)-module struc-ture, given by the right multiplication of derivatives. The inclusion mapS(Gw, V )′ ↪→ S(G≥w, V )′ is a right U(p∅)-homomorphism.Remark 7.7. Starting from this subsection, through the entire chapter, wewill consider the Ker(resw) and S(G≥w, V )′ as a right U(g)-module, andsimilarly S(Gw, V )′ as a right U(p∅)-module, with the transpose right modulestructures from the original left module structures on them.We will study the right n∅-torsion subspace[Ker(resw)][n∅•]which is exactly the left torsion subspace [Ker(resw)][n∅•] by Lemma 7.6.1687.1.3 Setting and Notation for the ProofRemember that in 7.1.2, we have regarded all modules as right modules.LetU = S(Gw, V )′K = Ker(resw)= Ker{S(G≥w, V )′ → S(G>w, V )′}= S(Gw, V )′ ⊗ U(n−w)= U ⊗ U(n−w)U[n∅•] = the n∅-torsion submodule of S(Gw, V )′M = U[n∅•] ⊗ U(n−w)The K is the space of (scalar valued) distributions on S(G≥w, V ) vanishingon the subspace S(G>w, V ), andM is the space of U(n−w)-transverse deriva-tives of (scalar valued) n∅-torsion distributions on S(Gw, V ). The M is asubspace of K, and we need to show it is exactly the n∅-torsion submoduleof K:K[n∅•] =M, (7.2)or equivalently (U ⊗ U(n−w))[n∅•] = U[n∅•] ⊗ U(n−w).For each n ∈ Z≥0, letUn(n−w) = spanC{X1 ·X2 · · ·Xk : k ≤ n,X1, . . . , Xk ∈ n−w}be the finite dimensional subspace of U(n−w), spanned by ≤ n products ofelements in n−w . Then {Un(n−w)}n≥0 form an exhaustive filtration of U(n−w):U(n−w) =⋃n≥0Un(n−w).LetKn = U ⊗ Un(n−w) (7.3)Kn[n∅•] = K[n∅•] ∩ KnMn = U[n∅•] ⊗ Un(n−w)Obviously we have Mn ⊂ Kn for all n ≥ 0, hence M ⊂ K. The {Kn}n≥0(respectively {Kn[n∅•]}n≥0, {Mn}n≥0) form an exhaustive filtration of K (re-spectively K[n∅•],M).To show K[n∅•] =M, we just need to show the two filtration {Kn[n∅•]}n≥0and {Mn}n≥0 are cofinal.1697.1.4 A FormulaIn this subsection, we show a multiplication formula in the envelopingalgebra. Namely, let g be an arbitrary complex Lie algebra, and U(g) be itsenveloping algebra, let Y1, . . . , Yk be k arbitrary elements in g and X ∈ gbe another one. We want to compute the product (Y1 · Y2 · . . . · Yk) ·X, andmove the Y1, . . . , Yk to the right as much as we can.Some NotationsFor an element X ∈ g, letRX : U(g)→ U(g), u 7→ u ·XLX : U(g)→ U(g), u 7→ X · uadX : U(g)→ U(g), u 7→ [X,u] = X · u− u ·Xbe the right multiplication, left multiplication, and adjoint action of X onU(g). In the algebra HomC(U(g), U(g)), these three elements satisfyadX = LX −RX . (7.4)The U(g) is an associative algebra, hence for X,Y ∈ g, we haveLX ◦RY = RY ◦ LX (7.5)i.e. the LX commutes with RY in HomC(U(g), U(g)) for all X,Y ∈ g,although the HomC(U(g), U(g)) is not a commutative algebra.For a positive integer k, we denote by[1, k] = {1, 2, . . . , k − 1, k}the set of the first k positive integers. For a subset S ⊂ [1, k], let s = |S| beits size, and Sc = [1, k] − S be its complement in [1, k]. For the subsets Sand Sc, we always arrange their elements in the increasing order.Let Y1, Y2, . . . , Yk ∈ g be k elements labeled by [1, k]. Let S = {i1, . . . , is}be a subset of [1, k] and let Sc = {j1, . . . , jk−s} be its complement. Weintroduce the following notations:〈YS ,−〉 = adYi1 ◦ adYi2 ◦ · · · ◦ adYisYS = Yi1 · Yi2 · · ·YisRYS = RYis ◦ · · · ◦RYi1170i.e. the 〈YS ,−〉 is the following element in the HomC(U(g), U(g))〈YS ,−〉 : U(g)→ U(g)u 7→ [Yi1 , [Yi2 , [· · · , [Yis , u]]]],the YS is the ordered product Y1 · · ·Ys in U(g), and the RYS is an elementin HomC(U(g), U(g)) which multiplies every element in U(g) by the productYS on the right.By convention, if S = ∅, we let YS = 1 ∈ U(g), and 〈YS ,−〉 = id andRYS = id in HomC(U(g), U(g)).The Attempts for Small k’sLet Y1, . . . , Yk be k arbitrary elements in g indexed by [1, k] and X ∈ gbe an arbitrary element. We want to write the productY1 · · ·Yk ·X = LY1 ◦ . . . ◦ LYk(X)in the form that Yi’s are on the right of X. We first look at the examples fork = 1, 2, 3, which inspire the general formula for all k. These are routine,but they help to understand the general formula.For k = 1, we see Y1 ·X = X · Y1 + [Y1, X]. In other word, we haveLY1(X) = RY1(X) + adY1(X) (7.6)as in (7.4).For k = 2, we have(Y1Y2) ·X = LY1 ◦ LY2(X)= LY1 ◦ (RY2 + adY2)(X) (by (7.6))= LY1 ◦RY2(X) + LY1 ◦ adY2(X)= RY2 ◦ LY1(X) + LY1 ◦ adY2(X) (by (7.5))= RY2 [RY1(X) + adY1(X)]+RY1 [adY2(X)] + adY1[adY2(X)] (by (7.6))i.e. in HomC(U(g), U(g)) we haveLY1LY2 = RY2RY1 +RY1adY2 +RY2adY1 + adY1adY2. (7.7)171For k = 3, we have(Y1Y2Y3) ·X = LY1LY2LY3(X)= LY1LY2 [RY3(X) + adY3(X)] (by (7.6))= LY1LY2RY3(X) + LY1LY2adY3(X)= RY3LY1LY2(X) + LY1LY2adY3(X) (by (7.5))= RY3RY2RY1(X) +RY3RY1adY2(X)+RY3RY2adY1(X) +RY3adY1adY2(X)+RY2RY1adY3(X) +RY1adY2adY3(X)+RY2adY1adY3(X) + adY1adY2adY3(X) (by (7.7))i.e. in HomC(U(g), U(g)), we haveLY1LY2LY3 = RY3RY2RY1 +RY3RY1adY2+RY3RY2adY1 +RY3adY1adY2+RY2RY1adY3 +RY1adY2adY3+RY2adY1adY3 + adY1adY2adY3 (7.8)The General FormulaWe now show the general formula:Lemma 7.8. For a positive integer k, and elements Y1, . . . , Yk, X ∈ g, wehave(Y1Y2 . . . Yk) ·X =∑S⊂[1,k]〈YS , X〉 · YSc (7.9)Equivalently in the algebra HomC(U(g), U(g)) we can writeLY1 ◦ . . . ◦ LYk(−) =∑S⊂[1,k](RYSc ◦ 〈YS ,−〉)=∑S⊂[1,k]〈YS ,−〉 · YScHere the sum on the right hand side has 2k terms when S runs through allsubsets of [1, k].Proof. As suggested by the cases of k = 1, 2, 3, we prove the (7.9) by in-duction on k. The case k = 1, 2, 3 are shown in the above subsubsection.Assume the (7.9) is true for k − 1 elements Y1, . . . , Yk−1.172For (Y1Y2 . . . Yk) ·X = LY1LY2 . . . LYk(X), we haveLY1LY2 . . . LYk(X) = LY1LY2 . . . LYk−1 [LYk(X)]= LY1LY2 . . . LYk−1 [RYk(X) + adYk(X)]= LY1LY2 . . . LYk−1RYk(X) + LY1LY2 . . . LYk−1adYk(X)For the first term, note that LY1 . . . LYk−1RYk = RYkLY1 . . . LYk−1 by(7.5), we haveLY1LY2 . . . LYk−1RYk(X) = RYkLY1LY2 . . . LYk−1(X)= RYk∑S⊂[1,k−1]〈YS , X〉 · YSc=∑S⊂[1,k−1](〈YS , X〉 · YSc) · Yk=∑S⊂[1,k]k∈Sc〈YS , X〉 · YScFor the second term, we haveLY1LY2 . . . LYk−1adYk(X) =∑S⊂[1,k−1]〈YS , adYk(X)〉 · YSc=∑S⊂[1,k−1]〈YS∪{k}, X〉 · YSc=∑S⊂[1,k]k∈S〈YS , X〉 · YScHence the sum of the above two terms is∑S⊂[1,k]k∈Sc〈YS , X〉 · YSc +∑S⊂[1,k]k∈S〈YS , X〉 · YSc =∑S⊂[1,k]〈YS , X〉 · YSci.e.LY1LY2 . . . LYk(X) =∑S⊂[1,k]〈YS , X〉 · YSc .By the same argument, or simply move the leading term in the sum tothe left, we have173Lemma 7.9. Let Y1, . . . , Yk, X ∈ g be arbitrary elements. ThenX · (Y1 · · ·Yk) =∑S⊂[1,k]YS · 〈X,YSc〉 (7.10)Here the last notation means〈X,YSc〉 = [· · · [X,Yj1 ] · · ·Yjk−s ]if Sc = {j1, . . . , jk−s}.7.1.5 Geometric PreparationWe study the tangent spaces on the regular submanifold Gw of Zw.Recall that Zw = PNPw is a Zariski open subset of G, and Gw = PwP∅ =PwN∅ = PwN+w is a Zariski closed subset of Zw. As manifolds, the Zw isan open submanifold of G and Gw is a regular closed submanifold of Zw.Let g0 be the (real) Lie algebra of G, and similarly for a subgroup ofG denoted by an uppercase letter, we use the same fraktur letter with asubscript 0 to denote its (real) Lie algebra.We regard the g0 as the abstract Lie algebra of G, and for an elementX ∈ g0, we denote by XL the corresponding left invariant vector field, andXLg ∈ TgG be the tangent vector of the vector field XL at g ∈ G. The XLand XLg are given by(XLf)(g) = XLg f =ddt|t=0f(getX), ∀f ∈ C∞(G).Similarly, we denote by XR the corresponding right invariant vector field onG and XRg ∈ TgG its tangent vector at g. The XR and XRg are given by(XRf)(g) = XRg f =ddt|t=0f(etXg), ∀f ∈ C∞(G).Remark 7.10. In most textbooks, the Lie algebra is defined to be the space{XL : X ∈ g0} of left invariant vector fields, and is canonically identifiedwith TeG = {XLe : X ∈ g0}. In this chapter, we will regard Lie algebraelements as vector fields (differential operators of degree 1), hence it is con-venient to distinguish vector fields from tangent vectors.We have the following easy facts in Lie theory:Lemma 7.11. For the two vector fields XL, XR corresponding to an elementX ∈ g0, they satisfy174• XLe = XRe , i.e. the two vector fields have the same tangent vector atthe identity.• XRg = (g−1Xg)Lg and XLg = (gXg−1)Rg , for all g ∈ G. (Here gXg−1means the adjoint action Adg(X).)The Tangent Space TwGwWe consider the tangent space TwGw of the submanifold Gw at w, wherew is a fixed representative inG (abuse of notation) of the Weyl group elementw. The TwGw is a subspace of TwG = TwZw, since Gw is a embeddedsubmanifold of Zw.At the point w ∈ Gw ⊂ G, the tangent space TwG = TwZw isTwG = TwZw = spanR{XLw : X ∈ g0} = spanR{XRw : X ∈ g0}.Recall that Nw = w−1NPw,N+w = Nw ∩ N∅, N−w = Nw ∩ N∅, andnw0, n+w0, n−w0 are their real Lie algebras respectively. Also the p0, nP0 arethe real Lie algebras of P,NP respectively. LetRp0w := spanR{XRw : X ∈ p0}Lnw0w := spanR{XLw : X ∈ nw0}Ln+w0w := spanR{XLw : X ∈ n+w0}Ln−w0w := spanR{XLw : X ∈ n−w0}e.g. Rp0w ⊂ TwG is the subspace of the TwG, spanned by tangent vectors XRwfor all X ∈ p0, etc. Then we have:Lemma 7.12. The above four spaces are subspaces of TwG = TwZw, and(1) Lnw0w = Ln+w0w ⊕ Ln−w0w .(2) TwG = TwZw = Rp0w ⊕ Lnw0w .(3) TwGw = Rp0w ⊕ Ln+w0w .(⊕ means direct sum of vector spaces)Proof. The (1) is trivial, since nw0 = n+w0 + n−w0.For the (2), first note that Rp0w ⊂ TwZw since Zw = PwNw is left P -stable, and Lnw0w ⊂ TwZw since Zw = PwNw is right Nw-stable (note Nw =w−1NPw). HenceTwG = TwZw ⊃ Rp0w + Lnw0w .175By counting the dimension, we just need to verify the two subspaces Rp0w andLnw0w are linear independent. Actually, let X ∈ nw0, then XLw = (wXw−1)Rw,and wXw−1 ∈ w(w−1nP0w)w−1 = nP0 which is linear independent from p0.Hence the tangent vector XLw = (wXw−1)Rw is linear independent with thesubspace Rp0w .For the (3), we also have TwGw ⊃ Rp0w + Ln+w0w since Gw = PwN+w is leftP -stable and right N+w -stable. Also by counting dimension, we just need toshow the two subspaces Rp0w , Ln+w0w are linear independent. This is obviousfrom (2) since Ln+w0w is a subspace of Lnw0w .The Tangent Spaces TpwnGwLet pwn (p ∈ P, n ∈ N+w ) be an arbitrary point in Gw (every element inGw is uniquely written as pwn for a pair of p ∈ P, n ∈ N+w ). We study thetangent space TpwnGw at pwn ∈ Gw.Similar to the tangent space at w, we have the following subspaces ofTpwnG = TpwnZwRp0pwn := spanR{XRpwn : X ∈ p0}Lnw0pwn := spanR{XLpwn : X ∈ nw0}Ln+w0pwn := spanR{XLpwn : X ∈ n+w0}Ln−w0pwn := spanR{XLpwn : X ∈ n−w0}i.e. Rp0pwn is the subspace of TpwnG = TpwnZw spanned by tangent vectors atpwn of right invariant vector fields in p0 etc, and all these four are subspacesof TpwnG = TpwnZw.As a generalization of the previous Lemma, we haveLemma 7.13. For arbitrary pwn ∈ Gw ⊂ Zw ⊂ G, the above four vectorspaces satisfy(1) Lnw0pwn = Ln+w0pwn ⊕ Ln−w0pwn.(2) TpwnG = TpwnZw = Rp0pwn ⊕ Lnw0pwn.(3) TpwnGw = Rp0pwn ⊕ Ln+w0pwn.(The Lemma 7.12 is the special case when p = n = e.)176Proof. (1)(3) are exactly the same as the previous lemma. The (2) is similar.Let X ∈ nw0, thenXLpwn = (pwnXn−1w−1p−1)Rpwn.Since n ∈ N+w ⊂ Nw, one has nXn−1 ∈ nw0 and wnXn−1w−1 ∈ nP0.Then since nP0 is linear independent with p0, we see pwnXn−1w−1p−1 ∈p(nP0)p−1 is linear independent with p(p0)p−1 = p0. Hence the subspaceLnw0pwn is linear independent with the subspace Rp0pwn.The n−w0 is Transverse to GwIn the last chapter, we call elements in U(n−w) “transverse derivatives”.We now explain this term.Definition 7.14. Let V be a set of vector fields on a manifold M , andN ⊂ M is a regular submanifold. Then we say V is transverse to thesubmanifold N , ifTxM = TxN ⊕ spanR{Xx : X ∈ V },for every x ∈ N , i.e. the vector fields in V give a linear complement of TxNin TxM at every point x of N .The Gw is a submanifold of Zw, with a smooth embedding i : Gw ↪→ Zw.Let TZw (resp. TGw) be the tangent bundle on Zw (resp. Gw), and leti∗TZw be the pull-back bundle over Gw. Then TGw is a subbundle of i∗TZw,and at every point pwn ∈ Gw, the fibre (tangent) space (TGw)pwn = TpwnGwis a subspace of (i∗TZw)pwn = TpwnZw.By the Lemma 7.13, we haveTpwnZw = TpwnGw ⊕ Ln−w0pwn(direct sum of vector spaces), i.e. the left invariant vector fields in n−w0 givea linear complement space of TpwnGw in TpwnZw at every point pwn in Gw.Hence the Lemma 7.13 tells usLemma 7.15. The subalgebra n−w0 (regarded as the space of left invariantvector fields) is transverse to the double coset Gw.177The Pull-Back Bundle i∗TZwLet i : Gw ↪→ Zw be the smooth embedding as above, and let i∗TZw bethe pull-back vector bundle of the tangent bundle TZw to Gw. The tangentbundle TGw of Gw is a subbundle of i∗TZw, and we have the following exactsequence of vector bundles:0→ TGw → i∗TZwNote that the tangent bundles TGw, TZw are algebraic vector bundles andthe map i is an algebraic morphism, hence the bundle i∗TZw and the abovebundle sequence are also algebraic.The total space of the bundle i∗TZw is{(x, v) : x ∈ Gw, v ∈ TxZw},and its fibre space (i∗TZw)x at a point x ∈ Gw is exactly the tangent spaceTxZw of Zw at x. Given an algebraic vector field V on Zw (a section ofthe bundle TZw), its restriction V |Gw to Gw is an algebraic section of thebundle i∗TZw.Algebraic Sections of the Bundle i∗TZwLet{X1, . . . , Xk} = a real basis of p0{Z1, . . . , Zl} = a real basis of n+w0{Y1, . . . , Yd} = a real basis of n−w0Let XRi be the right invariant vector fields on Zw (or G) corresponding toXi for all i = 1, . . . , k. Similarly, let ZLi , i = 1, . . . , l and YLj , j = 1, . . . , dbe the left invariant vector fields corresponding to Zi, Yj respectively. InLemma 7.13, we have seen the{XRi,x, . . . , XRk,x} ∪ {ZL1,x, . . . , ZLl,x} ∪ {Y L1,x, . . . , Y Ld,x}form a basis of the fibre space (i∗TZw)x = TxZw of i∗TZw for every pointx ∈ Gw, and the subset{XR1,x, . . . , XRk,x} ∪ {ZL1,x, . . . , ZLl,x}is a basis of the subspace TxGw for every x ∈ Gw.178Lemma 7.16. Under the above basis, every algebraic section s of the bundlei∗TZw on Gw is of the forms(x) =k∑i=1Ai(x)XRi,x +l∑i=1Bi(x)ZLi,x +d∑i=1Ci(x)Y Li,xfor a unique set of regular functions Ai, i = 1, . . . , k, Bi, i = 1, . . . , l andCi, i = 1, . . . , d. (If we write x = pwn for unique p ∈ P, n ∈ Nw, theAi, Bi, Ci are regular functions of p and n.)The Coefficients Ai, Bi, CiLet H ∈ g0 be an arbitrary element. For each n ∈ N+w , consider theconjugation wnHn−1w−1 ∈ g0. Since g0 = p0 + nP0, the wnHn−1w−1 isuniquely written aswnHn−1w−1 = Hp0 +HnP0where Hp0 ∈ p0, HnP0 ∈ nP0. Therefore the H decomposes asH = n−1w−1Hp0wn+ n−1w−1HnP0wn.Note that n−1w−1HnP0wn is in n−1w−1nP0wn = n−1nw0n = nw0 (sincen ∈ N+w ⊂ Nw). Hence we can further decompose it asn−1w−1HnP0wn = H+ +H−where H+ ∈ n+w0, H− ∈ n−w0. We denote byH ′ = n−1w−1Hp0wn.Then H is a sum of the formH = H ′ +H+ +H−where H ′ ∈ n−1w−1p0wn, H+ ∈ n+w0, H− ∈ n−w0. Note that the three com-ponents H ′, H+, H− depend on n, hence one has different H ′, H+, H− atdifferent point n ∈ N+w .Now let pwn ∈ Gw be an arbitrary point (the p ∈ P, n ∈ N+w are uniquelydetermined by the point), and H ∈ g0 be an arbitrary element in the realLie algebra. For the n ∈ N+w , let H ′, H+, H− be three components of H179as above, and let (H ′)L, (H+)L, (H−)L be the corresponding left invariantvector fields, thenHL = (H ′)L + (H+)L + (H−)L.At the point pwn ∈ Gw, the tangent vector HLpwn decomposes intoHLpwn = (H′)Lpwn + (H+)Lpwn + (H−)Lpwn (7.11)= (pwnH ′n−1w−1p−1)Rpwn + (H+)Lpwn + (H−)LpwnSince H+ ∈ n+w0, H− ∈ n−w0, one has (H+)Lpwn ∈ Ln+w0pwn and (H−)Lpwn ∈ Ln−w0pwn.Also by H ′ = n−1w−1Hp0wn ∈ n−1w−1p0wn, one haspwnH ′n−1w−1p−1 = pHp0p−1 ∈ p0,hence (H ′)Lpwn = (pwnH ′n−1w−1p−1)Rpwn ∈ Rp0pwn.Therefore the above decomposition (7.11) agrees with the tangent spacedecomposition in Lemma 7.13:TpwnZw = Rp0pwn ⊕ Ln+w0pwn ⊕ Ln−w0pwn.In sum, given an element H ∈ g0, and let HL be the corresponding leftinvariant vector field on G. If we restrict HL to the open submanifold Zw,we get an algebraic section in Γ(Zw, TZw), and by abuse of notation, westill denote it by HL. The restriction HL|Gw of HL ∈ Γ(Zw, TZw) to Gwis an algebraic section in Γ(Gw, i∗TZw), and by Lemma 7.16, the HL|Gw isuniquely written as(HL|Gw)(pwn) = HLpwn (7.12)=k∑i=1Ai(pwn)XRi,pwn +l∑i=1Bi(pwn)ZLi,pwn+d∑i=1Ci(pwn)Y Li,pwn, (7.13)where Xi, Zi, Yi are the basis of subalgebras as above, and Ai, Bi, Ci arealgebraic functions on Gw (of p and n).Remark 7.17. Given an H ∈ g0, we can find the coefficient functionsCi(pwn) in the following way:180(1) conjugate H to nHn−1;(2) decompose the nHn−1 into a sumnHn−1 = (nHn−1)w−1p0w + (nHn−1)w−1nP0w(3) conjugate the second component (nHn−1)w−1nP0w ∈ w−1nP0w back ton−1(nHn−1)w−1nP0wn(4) then the Ci(pwn) is the coefficient of the base vector Y ipwn in(n−1(nHn−1)w−1nP0wn)Lpwn7.2 The Inclusion M⊂ K[n∅•]In this section, we show the inclusion M⊂ K[n∅•] by showingMk ⊂ K[n∅•]for all k ≥ 0, since the {Mk}k≥0 form an exhaustive filtration of M.The distributions in K = U ⊗ U(n−w) are transverse derivatives of distri-butions on Gw, while the n∅ acts on such distributions as derivatives. LetΦ ∈ U = S(Gw, V )′ and u ∈ Un(n−w), then the Φ ⊗ u = Φ · u is a typicalelement in K. Let u′ ∈ U(n∅), then the right multiplication is given by(Φ · u) · u′ = Φ · (u · u′)where u · u′ is the multiplication in U(g) or in the larger algebra D(G≥w) ofalgebraic differential operators on G≥w.The elements in U(n−w) “protects” the elements in U from elimination byU(n∅). The essence of inclusion M ⊂ K[n∅•] is: the U(n∅) can always “peeloff” the protection from U(n−w), and annihilate the inner distributions in U ,if they are n∅-torsion.The main idea is to use induction, and show one can peel off “one layerof the U(n−w)” at a time. Then after sufficiently many steps, the U(n∅) canreach the U part of the distributions.7.2.1 Some Algebraic Lemmas—IWe show some algebraic results before proving the inclusionM⊂ K[n∅•].181A Product FormulaLet h be an arbitrary Lie algebra.Lemma 7.18. For arbitrary n ∈ Z>0, let H,X1, . . . , Xn ∈ h be arbitraryelements. Then in U(h) we have the formula:X1 ·X2 · · ·Xn ·H = H ·X1 · · ·Xn −n∑i=1X1 · · ·Xi−1 · [H,Xi] ·Xi+1 · · ·Xn.(7.14)Proof. We prove the formula by induction on n. For n = 1, we haveX1 ·H = H ·X1 − [H,X1],which verifies (7.14).Suppose (7.14) is true for n = k, then for n = k + 1, we haveX1 · · ·Xk ·Xk+1 ·H = X1 · · ·Xk · (Xk+1 ·H)= X1 ·Xk · (H ·Xk+1 − [H,Xk+1])= (X1 · · ·Xk−1 ·Xk ·H) ·Xk+1 −X1 · · ·Xk · [H,Xk+1]= (H ·X1 · · ·Xk −k∑i=1X1 · · · [H,Xi] · · ·Xk) ·Xk+1−X1 · · ·Xk · [H,Xk+1]= H ·X1 · · ·Xk+1 −k+1∑i=1X1 · · · [H,Xi] · · ·Xk+1This lemma is easier to prove than Lemma 7.8, hence it is less powerfulthan Lemma 7.8. The advantage of Lemma 7.8 is: all Yi’s are on the rightend of the product.Torsion Submodule is StableLet h be an arbitrary complex Lie algebra, and h1, h2 be two subalgebrasof h such thath = h1 + h2i.e. h is the direct sum of h1, h2 as vector spaces. Suppose h1 normalizes h2:[h1, h2] ⊂ h2,or equivalently h2 is an ideal of h. We have182Lemma 7.19. LetU = a right h-module, also a right U(h)-module,and we can regard it as a h2-moduleU[h•2] = the h2-torsion subspace of UThen each h2-annihilator U[hk2 ]is a h-submodule of U . In particular, theh2-torsion subspace U[h•2] is a h-submodule of U .Proof. We just need to show each U[hk2 ]is stable under the left h-action.Obviously it is stable under h2-action, hence we just need to show it isstable under h1-action.Let k ∈ Z>0 be arbitrary (when k = 0, U[h02] = U , there is nothing to beproved). Let u ∈ U[hk2 ] be an arbitrary element, i.e. for any X1, . . . , Xk ∈ h2,one hasu ·X1 · · ·Xk = 0.Let H ∈ h1 be an arbitrary element, we show u · H ∈ U[hk2 ], i.e. it isannihilated by products of k elements in h2. For arbitrary X1, . . . , Xk ∈ h2,by (7.14), we haveu ·H ·X1 · · ·Xk = u · (H ·X1 · · ·Xk)= u ·X1 · · ·Xk ·H +k∑i=1u ·X1 · · · [H,Xi] · · ·XkThe first term is zero since u ·X1 · · ·Xk = 0. Every term in the second sumis zero, since [H,Xi] ∈ h2. (Note that we assumed h2 to be normalized byh1.) Hence u ·H ·X1 · · ·Xk = 0 for all X1, . . . , Xk ∈ h2 and H ∈ h1.Torsion Submodule is AbsorbingLemma 7.20. Leth = a complex Lie algebraU = a right h-moduleU[h•] = the h-torsion submodule of ULet u ∈ U be an element, and assume there exists an integer n > 0 such thatu ·X1 · · ·Xn ∈ U[h•]for all X1, . . . , Xn ∈ h. Then u ∈ U[h•].183Proof. Let u and n be as in the Lemma. Let b1, . . . , bd be a basis of h.Note that the n is “uniform” for arbitrary X1, . . . , Xn. In particular wecan pick Xi’s from the basis {b1, . . . , bd}, and there are dn choices. Thedn elements of the form X1 · X2 · · ·Xn (each Xi is from the above basis)generate the ideal (hn).For each product X1 · · ·Xn (Xi from the above basis), the u ·X1 · · ·Xnis in U[h•], hence(u ·X1 · · ·Xn) · (hm) = 0for a m > 0 depending on the choices of Xi. Since there are only dn choicesof the product X1 · · ·Xn, we pick the largest m and denote it by M , then(u ·X1 · · ·Xn) · (hM ) = 0for all dn choices of X1, . . . , Xn. Hence u · (hn+M ) = 0 and u ∈ U[h•].The Quotient g/p∅ is a n∅-Torsion ModuleThe adjoint representation of n∅ on g makes g into a (finite dimensional)n∅-torsion module. The p∅ ⊂ g is a n∅-submodule (parabolic subalgebras areself-normalizing), hence the quotient module g/p∅ is a n∅-torsion module.Hence we haveLemma 7.21. Let Y ∈ g be an arbitrary element. Then there exists apositive integer n (depending only on Y ), such that[· · · [[Y,X1], X2] · · ·Xn] ∈ p∅,for all X1, . . . , Xn ∈ n∅.7.2.2 The Inclusion Mk ⊂ K[n∅•]We show the following Lemma by induction on k:Lemma 7.22. For each k ≥ 0, the subspaceMk = U[n∅•] ⊗ Uk(n−w)is contained in K[n∅•].The case k = 0 is trivial. Namely theM0 = U[n∅•] ⊗ C = U[n∅•] = S(Gw, V )′[n∅•]184is contained in K[n∅•], since the inclusion U ↪→ K is a right n∅-homomorphismand the n∅-torsion functor is left exact (Lemma 3.36).(Induction Hypothesis): We assumeMk−1 ⊂ K[n∅•].We show the Mk is contained in K[n∅•].Proof. By the Poincare-Birkhoff-Witt theorem, every element in Mk is asum of terms of the following form:Φ⊗ Y1 · Y2 · · ·Yi = Φ · Y1 · · ·Yiwhere Φ ∈ U[n∅•], 0 ≤ i ≤ k and Y1, . . . , Yi ∈ n−w . If i < k, then by inductionhypothesis, such elements are already contained in K[n∅•], since they are inMi ⊂ Mk−1. We just need to show elements of the following form arecontained in K[n∅•]:Φ · Y1 · · ·Yk,where Φ ∈ U[n∅•] and Y1, . . . , Yk ∈ n−w .First we writeΦ · Y1 · · ·Yk = (Φ · Y1 · · ·Yk−1) · Yk.The Φ · Y1 · · ·Yk−1 is in Mk−1 and by the induction hypothesis, it is n∅-torsion, hence there exists an integer n1 > 0 such that(Φ · Y1 · · ·Yk−1) · (n∅n1) = {0}.By Lemma 7.21, there exists an integer n2 > 0 such that[· · · [[Yk, X1], X2] · · · ], Xn2 ] ∈ p∅,for all X1, . . . , Xn2 ∈ n∅.Now let n = n1 + n2 + 1, then for any n elements X1, . . . , Xn ∈ n∅, wehave (by (7.10))(Φ · Y1 · · ·Yk) ·X1 · · ·Xn = (Φ · Y1 · · ·Yk−1) · (Yk ·X1 · · ·Xn)=∑S⊂[1,n](Φ · Y1 · · ·Yk−1) ·XS · 〈Yk, XSc〉If |S| ≥ n1, then (Φ · Y1 · · ·Yk−1) · XS = 0, since XS ∈ (n∅n1) and wehave (Φ · Y1 · · ·Yk−1) · (n∅n1) = {0} as above.185If |Sc| ≥ n2, then 〈Yk, XSc〉 ∈ p∅. By the induction hypothesis, theΦ · Y1 · · ·Yk−1 ∈ K[n∅•], hence (Φ · Y1 · · ·Yk−1) · XS ∈ K[n∅•], since K[n∅•] isa n∅-submodule of K and is stable under multiplication of U(n∅). Then byapplying Lemma 7.19 to h = p∅, h1 = m∅, h2 = n∅, we know the K[n∅•] isp∅-stable, and(Φ · Y1 · · ·Yk−1) ·XS · 〈Yk, XSc〉 ∈ K[n∅•].Since we have chosen n = n1 + n2 + 1, then either |S| ≥ n1 or |Sc| ≥ n2.Hence the term(Φ · Y1 · · ·Yk−1) ·XS · 〈Yk, XSc〉is either zero, or in K[n∅•]. In sum, the summands (Φ · Y1 · · ·Yk−1) · XS ·〈Yk, XSc〉 are all in K[n∅•], hence(Φ · Y1 · · ·Yk) ·X1 · · ·Xn ∈ K[n∅•]for all X1, . . . , Xn ∈ n∅. Then by Lemma 7.20, we seeΦ · Y1 · · ·Yk ∈ K[n∅•].7.3 The Inclusion K[n∅•] ⊂MSimilar to the proof of Lemma 3.52, the crucial idea to prove(U ⊗ U(n−w))[n∅•] ⊂ U[n∅•] ⊗ U(n−w)is to find a “good basis” of the enveloping algebra U(n−w).Let Y1, . . . , Yd be a set of basis of the Lie algebra n−w . Let I = (i1, . . . , id)be a multi-index in Zd≥0, and we denote byY I = Y i11 · Y i22 · · ·Y iddthe ordered product. Then the P-B-W Theorem tells us the {Y I : I ∈ Zd≥0}form a basis of the enveloping algebra U(n−w).Therefore every element in K is uniquely written as a finite sum∑I∈Zd≥0ΦI ⊗ Y I =∑I∈Zd≥0ΦI · Y I186where ΦI ∈ U and all but finitely many terms in the sum are zero. Thiselement (sum) in K is in M if and only if each nonzero ΦI is in U[n∅•].As an analogue to the Lemma 3.52, the main difficulty of the proof is: ifthe above sum is n∅-torsion, one cannot see whether each summand ΦI · Y Iis n∅-torsion. The idea to prove the inclusion is very similar: we need tosingle out one summand at a time and show it is torsion.Unlike the proof of Lemma 3.52, the tensor product U ⊗ U(n−w) is not atensor product of n∅-modules, and the multiplication of n∅ on it is extremelycomplicated, such that one cannot expect a simple multiplication formulaas (3.2).This requires us to arrange the PBW-basis Y I by a linear order on themulti-index set, so that we can perform induction on this linear order toseparate the summands. This order is constructed by the following twosteps:• First fix a basis of n−w consists of root vectors, and arrange them asY1, . . . , Yd (here d is the dimension of n−w) such that their “height” arenon-decreasing (see 7.3.3).• Second we define a “linear order” on the index set Ld = Zd≥0 (see 7.3.1).The the above fixed basis {Y1, . . . , Yd} determines a linear order on thePBW-basis{Y I = Y i11 Y i22 · · ·Y idd : I = (i1, . . . , id) ∈ Ld = Zd≥0}.Combining the above two aspects, we have a good linear order “≤” onthe PBW-basis {Y I : I ∈ Ld}. LetUI(n−w) = the subspace spanned by YJ for all J ≤ I.Then we define a filtration on the K and its subspace K[n∅•] byKI := U ⊗ UI(n−w)KI[n∅•] := KI ∩ K[n∅•]The {KI[n∅•] : I ∈ Ld} forms an exhaustive filtration of K[n∅•], and we justneed to show each subspace KI[n∅•] is included in M.The key points of the proof are• Each subspace KI[n∅•] in the filtration is n∅-stable (Lemma 7.44), henceis a n∅-submodule of K.• Each graded piece of the filtration {KI[n∅•] : I ∈ Ld} is isomorphic toU as n∅-modules.1877.3.1 Orders on Multi-index SetsFor a positive integer d, letLd = Zd≥0 = {(i1, . . . , id) : i1, . . . , id ∈ Z≥0}be the set of multi-indices with d components of non-negative integers. TheLd is an additive semigroup. We denote a generic element in Ld by I =(i1, . . . , id), and we denote the sum of its components by |I| = i1 + . . .+ id.For each m ∈ Z≥0, we letLdm = {I ∈ Ld : |I| = m} = {(i1, . . . , id) : i1 + . . .+ id = m}be the subset of multi-indices with sums of their components equal to m.Then Ld =⋃m≥0 Ldm is a disjoint union. It is easy to count the size of Ldm:|Ldm| = Cmd+m−1.For each m ∈ Z≥0, we letLd≤m =⋃0≤k≤mLdk.Then Ld =⋃m≥0 Ld≤m.The Reverse Lexicographic Order on LdmLet Ldm = {I ∈ Ld : |I| = m} be as above. We define an order on Ldm.Definition 7.23 (The Reverse Lexicographic Order on Ldm). Let I =(i1, . . . , id), J = (j1, . . . , jd) be two elements in Ldm. We defineI < Jif there is an l, such that 0 ≤ l ≤ d andid = jd, id−1 = jd−1, . . . , il+1 = jl+1, il < jl.This is an order called the reverse lexicographic order on Ldm. Wedenote by I ≤ J if I < J or I = J .Remark 7.24. It is easy to see this order is a linear order (total order), andis a well-order since Ldm is finite and discrete. Actually the above reverselexicographic order is defined on the entire Ldm, and we just restrict it toeach subset Ldm.188Example 7.25. Consider the case d = 3,m = 4. It contains C46 =6!4!2! = 15elements, and they are ordered as the following linear sequence:(0, 0, 4) > (0, 1, 3) > (1, 0, 3)> (0, 2, 2) > (1, 1, 2) > (2, 0, 2)> (0, 3, 1) > (1, 2, 1) > (2, 1, 1) > (3, 0, 1)> (0, 4, 0) > (1, 3, 0) > (2, 2, 0) > (3, 1, 0)> (4, 0, 0)The Linear Order on LdSince the Ld is the disjoint union of Ldm,m ≥ 0, we can connect the linearorders on each Ldm to obtain a linear order on Ld.Definition 7.26 (linear order on Ld). Let I = (i1, . . . , id), J = (j1, . . . , jd)be two elements in Ld. We defineI < Jif |I| < |J | or |I| = |J | = m and I < J under the reverse lexicographic orderon Ldm. We call this order the linear order on Ld.Example 7.27. For d = 2, the linear order on L2 is just the snake-likeorder parameterizing the rational numbers.Remark 7.28. Note that the above linear order on Ld is NOT the reverselexicographic order. For example, let d = 2, then under the lexicographicorder, one has (3, 0) < (1, 1) by comparing the second component. Howeverunder our order, the (3, 0) > (1, 1) since the sum 3 + 0 > 1 + 1.Why Linear Order?Why not use the reverse lexicographic order on Ld? Because we cannotperform induction on the reverse lexicographic order.In Figure 7.2, we show the linear and reverse lexicographic order for L2..In both figures, each dot is an element in L2, and each dot has an arrowpointing to the adjacent element which is larger than it in the correspondingorder, i.e. one has an increasing sequence following the arrows.As we can see from the Figure 7.2 (a), each dot has a unique adjacent dot“smaller“ than it. Starting from an arbitrary dot, one can always descendto the initial dot (0, 0) after finitely many steps.189i1i2(a) Linear order d = 2i1i2(b) Reverse lexicographic order d = 2Figure 7.2: Linear order and Reverse lexicographic order (d=2)However in Figure 7.2 (b), one can see the dot (0, 1) has no adjacentelement smaller than it. In other word, under the reverse lexicographic190order, the subset {I ∈ L2 : I < (0, 1)} contains all (i, 0), i ≥ 0, and thissubset has no maximal element. Hence starting from (0, 1), one cannotdescend to the initial point (0, 0) after finite steps, therefore one cannotperform induction on this order.A Partial Order on LdWe define a partial order on the LdDefinition 7.29. Let I = (i1, . . . , id), J = (j1, . . . , jd) be two elements inLd. We defineI Jif i1 ≤ j1, i2 ≤ j2, . . . , id ≤ jd. And we denote by I ≺ J if I J but I 6= J .It is easy to see the is a partial order on Ld, and we call it the componentorder on Ld.Lemma 7.30. For two I, J ∈ Ld, if I ≺ J , then |I| < |J | hence I < Junder the linear order on Ld.Definition 7.31. Let I = (i1, . . . , id), J = (j1, . . . , jd) and suppose I J .We define their difference byJ − I := (j1 − i1, . . . , jd − id).Obviously we have J − I ∈ Ld, and J − I J and |J − I| = |J | − |I|.The Fortified Formula of (7.9) and (7.10)Let g be a complex Lie algebra, and Y1, . . . , Yd be d arbitrary elementsin g. For a multi-index I = (i1, . . . , id) ∈ Ld, we denote byY I = Y i11 Yi22 · · ·Y idd .And we denote by 〈Y I ,−〉 the following element in HomC(U(g), U(g))〈Y I , u〉 := (adY1)i1 ◦ (adY2)i2 ◦ . . . ◦ (adYd)id(u),for all u ∈ U(g), and 〈−, Y I〉 the following element in HomC(U(g), U(g)):〈u, Y I〉 := (−1)|I|(adYd)id ◦ . . . ◦ (adY1)i1(u).By convention, if I = (0, . . . , 0), then 〈Y I ,−〉 = 〈−, Y I〉 = id.With the above partial order on Ld and notations, we have the fortifiedversion of the formulas (7.9) and (7.10).191Lemma 7.32. Let X ∈ g be an arbitrary element. Then we have the fortifiedformula of (7.9):Y I ·X =∑JICJI 〈Y J , X〉 · Y I−J (7.15)=∑j1≤i1,...,jd≤idCJI 〈Y J , X〉 · Y i1−j11 · · ·Y id−jdd .and the fortified formula of (7.10):X · Y I =∑JICJI YJ · 〈X,Y I−J〉. (7.16)Here the coefficient CJI is the product of combinatoric coefficients:CJI =(j1i1)·(j2i2)· · ·(jdid).Proof. To show (7.15) we simply use the (7.9). By counting the multiplicitiesof repeating terms, we have the combinatoric coefficients in (7.15). The(7.16) is shown in the same way.7.3.2 Some Algebraic Lemmas—IIThe Poincare-Birkhoff-Witt BasisLet h be an arbitrary complex Lie algebra, with dimension d, and letY1, . . . , Yd be an arbitrary basis of h. For a multi-index I = (i1, . . . , id) ∈Ld = Zd≥0, we letY I = Y i11 · · ·Y idd ,then {Y I : I ∈ Ld} form a basis of the enveloping algebra U(h), called aPoincare-Birkhoff-Witt basis of h, or simply a PBW basis.Lemma 7.33. Let I ∈ Ld be an arbitrary multi-index, and let Y I be thePBW basis element as above. Let Y ∈ h be an arbitrary element, then theY · Y I is contained in the linear span of{Y J : |J | ≤ |I|+ 1}.Equivalently, this meansh · Un(h) ⊂ Un+1(h).192Proof. We just need to show the case when Y = Yk is a basis vector from{Y1, . . . , Yd}. We prove by induction on |I|. If |I| = 0, then Y I = 1 andthe result is trivial. If |I| = 1, then Y I is a single basis vector Yl for some1 ≤ l ≤ d. ThenYkYl =YkYl, if k < lY 2l , if k = lYlYk + [Yk, Yl], if k > land the result is true.For a general Y I , we haveYk · Y I = Yk · Y i11 · · ·Y idd= (YkY1)Yi1−11 Yi22 · · ·Y idd= Y1(YkYi1−11 Yi22 · · ·Y idd ) + [Yk, Y1]Y i1−11 Y i22 · · ·Y iddBy the induction hypothesis, the YkYi1−11 Yi22 · · ·Y idd is in the span of {Y J :|J | ≤ |I| − 1 + 1} and the [Yk, Y1]Y i1−11 Y i22 · · ·Y idd is in the span of {Y J :|J | ≤ |I|−1+1}, hence the Yk ·Y I is in the span of {Y J : |J | ≤ |I|+1}.By iteration, i.e. by repeatedly using the above Lemma, we haveLemma 7.34. Let X1, . . . , Xk be arbitrary element in h, and I, YI as above.Then theX1 · · ·Xk · Y Iis a linear combination of Y J such that |J | ≤ |I|+ k.A Formula on Lie Brackets in Enveloping AlgebraLet h be an arbitrary complex Lie algebra, and U(h) be its envelopingalgebra.Lemma 7.35. Let X,Y1, . . . , Yk be arbitrary elements in h. Then in U(h)we have[X,Y1 · · ·Yk] =k∑i=1Y1 · · ·Yi−1[X,Yi]Yi+1 · · ·Yk (7.17)and similarly[Y1 · · ·Yk, X] =k∑i=1Y1 · · ·Yi−1[Yi, X]Yi+1 · · ·Yk (7.18)193Proof. We prove the first one by induction on k, and the second equality isimplied immediately by the first one.For k = 1, the equality is trivial. We assume the equality holds for k−1,then[X,Y1 · · ·Yk] = XY1 · · ·Yk − Y1 · · ·YkX= XY1Y2 · · ·Yk − Y1XY2 · · ·Yk + Y1XY2 · · ·Yk − Y1 · · ·YkX= [X,Y1]Y2 · · ·Yk + Y1(XY2 · · ·Yk − Y2 · · ·YkX)By the induction hypothesis, we haveXY2 · · ·Yk − Y2 · · ·YkX = [X,Y2 · · ·Yk]=k∑i=2Y2 · · ·Yi−1[X,Yi]Yi+1 · · ·YkHence[X,Y1 · · ·Yk] = [X,Y1]Y2 · · ·Yk + Y1k∑i=2Y2 · · ·Yi−1[X,Yi]Yi+1 · · ·Yk=k∑i=1Y1 · · ·Yi−1[X,Yi]Yi+1 · · ·YkBy a straightforward application of the above Lemma, we haveLemma 7.36. Let I = (i1, . . . , id) ∈ Ld be a multi-index, and X,Y1, . . . , Ydbe arbitrary elements in h, and let Y I = Y i11 · · ·Y idd be the ordered product.Then[X,Y I ] =d∑k=1ik−1∑l=0Y i11 · · ·Y ik−1k−1 Y lk [X,Yk]Y ik−l−1k Yik+1k+1 · · ·Y idd (7.19)and similarly[Y I , X] =d∑k=1ik−1∑l=0Y i11 · · ·Y ik−1k−1 Y lk [Yk, X]Y ik−l−1k Yik+1k+1 · · ·Y idd (7.20)1947.3.3 Heights on Basis of n−wThe real Lie algebra n−w0 = w−1nP0w ∩ n∅0 of the real group N−w =w−1NPw∩N∅ is a direct sum of restricted (relative) root spaces. In partic-ular, we can find a R-basis of n−w0 (which is also a C-basis of the complexifiedLie algebra n−w) consisting of root vectors.The roots occurring in n−w (or n−w0) form the subset Σ−∩w−1(Σ−−Σ−Θ),and this subset is closed under addition. For a root α occurring in n−w , letg0α be the root space, its complexification gα is exactly the relative rootspace. The g0α is an abelian subalgebra of g0 if 2α is not a root. (Note thatits dimension may not be one since the group G may not be split.)Let ∆ = {α1, . . . , αr} be the fixed simple system of the relative (re-stricted) root system. Since Σ− ∩ w−1(Σ− − Σ−Θ) is contained in Σ−, everyroot in it is uniquely written as an integral combination of roots in −∆. Letα ∈ Σ− ∩ w−1(Σ− − Σ−Θ), and suppose it is of the formα = −∑niαifor ni ∈ Z≥0. We define the height of α to beHt(α) =r∑i=1ni.Remark 7.37. Note that this is not the ordinary height function in thetheory of root system, since we are working with negative roots. The above“height” Ht is the height function under the base −∆ in the ordinary sense.Ordered Basis of n−wFrom this section through the entire chapter, we fix a R-basis of n−w0consisting of root vectors, and they are also C-basis of the complexified Liealgebra n−w .Let d be the dimension of the n−w , and we label the basis of root vectors asY1, . . . , Yd, and define the height of Yi to be the height of the correspondingroot. We label the Yi’s such thatHt(Y1) ≤ Ht(Y2) ≤ . . . ≤ Ht(Yd),i.e. the height of the corresponding roots of Yi increase with i.195Lemma 7.38. For two basis root vectors Yi, Yj (i ≤ j), we have[Yi, Yj ] =∑k≥jHt(Yk)>Ht(Yj)ckYk, (7.21)i.e. the [Yi, Yj ] is a linear combination of basis vectors with strictly largerheights.In particular, the basis root vectors corresponding to the highest roots(e.g. the Yd) are in the center of n−w .Proof. For two roots α, β occurring in n−w , we have[g0α, g0β] ⊂{g0(α+β), if α+ β ∈ Σ− ∩ w−1(Σ− − Σ−Θ){0}, otherwise.The Lemma is implied immediately by this fact.7.3.4 The Coefficients CiLet Yi be a base vector in the fixed basis {Y1, . . . , Yd} of n−w0 and supposewe label the basis such that their height are increasing. Without loss ofgenerality, let X ∈ n∅0 be a root vector in the real Lie algebra n∅0. Inthis subsection, we study the element [Yi, X] ∈ g0, and its decompositioninto linear combination of base vector fields (with coefficients in the ring ofalgebraic functions).Remark 7.39. Note that [Yi, X] may not be in p∅0 or n−w0 (neither tangentnor transverse), and it could be an element in w−1p0w ∩ n∅0.Lemma 7.40. Let Ck be the coefficient function of Y Lk in the expression(7.11) of [Yi, X], then we have Ck = 0 for all k ≥ i.Proof. By the assumption, both Yi and X are root vector, we may assumethat [Yi, X] is also a root vector. We go through the 4 steps in Remark 7.17to compute Ck.In the step (1) of Remark 7.17, we may assume n = exp(X) for someX ∈ n+w0, then by the Baker-Campbell-Hausdorff formula, we havenHn−1 = H + [X,H] +12![X, [X,H]] + . . .On the right-hand-side, every term has strictly lower heights than Yi.196According to the step (2) of Remark 7.17, we need to decompose thenHn−1 into two componentsnHn−1 = (nHn−1)w−1p0w + (nHn−1)w−1nP0w.By the step (1) above, we see the second component (nHn−1)w−1nP0w is alinear combination of root vectors with strictly lower heights than Yi. (Thecoefficients are functions of n)According to the step (3) of Remark 7.17, if we conjugate the secondcomponent back ton−1(nHn−1)w−1nP0wnagain by Baker-Campbell-Hausdorff, it is still a linear combination of rootvectors with strictly lower heights than Yi.Therefore, all root vectors Yk, k ≥ i cannot occur in the linear combina-tion (i.e. coefficients are zero), hence Ck(pwn) = 0 for all k ≥ i.7.3.5 The Elements Yk · Y ILet Y1, . . . , Yd be a basis of n−w consisting of root vectors and satisfyingHt(Y1) ≤ Ht(Y2) ≤ . . . ≤ Ht(Yd).For a multi-index I = (i1, . . . , id) ∈ Ld, we denote by Y I the ordered productY i11 · · ·Y idd , and they form the PBW basis of U(n−w).Let Yk be an arbitrary basis vector from {Y1, . . . , Yd}, we consider theelement Yk · Y I . It is hard to find a expression of it as a combination ofPBW-basis. However we can estimate the PBW-basis vectors Y J occurringin the expression of Yk ·Y I , and have the following result which is a fortifiedversion of Lemma 7.33:Lemma 7.41. For all Yk, 1 ≤ k ≤ d, the Yk · Y I is a linear combination ofY J such thatJ ≤ (i1, . . . , ik−1, ik + 1, ik+1, . . . , id).(under the linear order on Ld). In particular, all such J satisfy |J | ≤ |I|+1,and this lemma implies the Lemma 7.33.The product YkYI = YkYi11 · · ·Y idd is in U(n−w) and is a linear combina-tion of PBW-basis Y J , J ∈ Ld. This Lemma estimate the “leading term” inthe expansion of YkYI as sum of PBW-basis. Note that the following proofis a proof by the first principal, not by induction.197Proof. We first look at some simple cases. If |I| = 0, or equivalently I =(0, . . . , 0), the statement in the Lemma is trivial.Next we look at the simplest nontrivial cases when |I| = 1, and assumeI = (0, . . . , 0, 1, 0, . . . , 0), i.e. the ith component is 1 and all other compo-nents are zero. Then Y I = Yi = Y1i , and we haveYkYI = YkYi =YkYi, if k < iY 2k = Y2i , if k = iYiYk + [Yk, Yi], if k > iThe cases k < i, k = i obviously verify the Lemma. The interesting case iswhen k > i, where a new term [Yk, Yi] is created. By Lemma 7.38, the [Yk, Yi]is a linear combination of root vectors with strictly larger heights, i.e. it isa combination of Yj such that j > i and j > k. However, such Yj ’s are stillsingle elements, and they are of the form Y J , where J = (0, . . . , 1, . . . , 0) (i.e.the jth component is 1 and all the other components are zero). Such J stillsatisfies J ≤ (i1, . . . , ik + 1, . . . , id), since |J | = 1. Hence the YiYk + [Yk, Yi]is still spanned by Y J with J ≤ (i1, . . . , ik + 1, . . . , id).Now we look at the general case Yk ·Y I , where I = (i1, . . . , id). We haveYk · Y I = Yk · Y i11 · · ·Y ik−1k−1 · Y ikk Yik+1k+1 · · ·Y idd= Y i11 · · ·Y ik−1k−1 · Y ik+1k · Yik+1k+1 · · ·Y idd+ [Yk, Yi11 · · ·Y ik−1k−1 ] · Y ikk Yik+1k+1 · · ·Y iddThe first term is Y J , where J = (i1, . . . , ik−1, ik + 1, ik+1, . . . , id), i.e. it isobtained from the original I by adding 1 to its kth component. This termhas J equals to the “upper bound” in the Lemma. We need to show thesecond term [Yk, Yi11 · · ·Y ik−1k−1 ] · Y ikk Yik+1k+1 · · ·Y idd is a linear combination ofY J such that J is strictly smaller than (i1, . . . , ik−1, ik + 1, ik+1, . . . , id).By (7.20) in Lemma 7.36, we have[Yk, Yi11 · · ·Y ik−1k−1 ] =k−1∑p=1ip−1∑l=0Y i11 · · ·Y lp [Yk, Yp]Y ip−l−1p Y ip+1p+1 · · ·Y ik−1k−1We just need to show for each p such that 1 ≤ p ≤ k− 1 and l such that0 ≤ l ≤ ip − 1, the productY i11 · · ·Y lp [Yk, Yp]Y ip−l−1p Y ip+1p+1 · · ·Y ik−1k−1 Y ikk Yik+1k+1 · · ·Y iddis a linear combination of Y J such that J < (i1, . . . , ik + 1, . . . , id).198We apply the Lemma 7.34 to the above elements. Note that the rearsegment Yip−l−1p Yip+1p+1 · · ·Y ik−1k−1 Y ikk Yik+1k+1 · · ·Y idd is of the form Y J where J =(0, . . . , 0, ip− l− 1, ip+1, . . . , ik, . . . , id), i.e. it is in the PBW-basis. And thefront segment Y i11 · · ·Y lp [Yk, Yp] is a (sum of) products of i1 + . . .+ip−1 + l+1elements. Then in Lemma 7.34, we see the elementY i11 · · ·Y lp [Yk, Yp]Y ip−l−1p Y ip+1p+1 · · ·Y ik−1k−1 Y ikk Yik+1k+1 · · ·Y iddis a combination of Y J where |J | ≤ (ip − l − 1 + ip+1 + . . . id) + (i1 + . . .+ip−1 + l + 1) = i1 + . . . + id = |I| < |I| + 1. Of course, these elements arelinear combination of Y J with J < (i1, . . . , ik+1, . . . , id) because the lengths|J | ≤ |I| < |I|+ 1.Remark 7.42. After proving this lemma, we found out this is a basic facton enveloping algebra. One don’t need the linear order on the PBW-basis,and one don’t need to label the basis of n−w according to the height either.7.3.6 The Submodule KILet d be the dimension of the Lie algebra n−w , and let Y1, . . . , Yd be areal basis of n−w0 (which is also a C-basis of n−w), and suppose we label themwith increasing heights: Ht(Y1) ≤ . . . ≤ Ht(Yd). As in the last subsection,for a multi-index I = (i1, . . . , id) ∈ Ld, we denote byY I = Y i11 · Y i22 · · ·Y iddthe corresponding monomial PBW-basis.Definition 7.43. For a multi-index I = (i1, . . . , id) ∈ Ld, the set {J ∈ Ld :J > I} has a unique minimal element under the linear order, and the (finite)set {J ∈ Ld : J < I} has a unique maximal element under the linear order.We denote them byI+ = the minimal element in {J ∈ Ld : J > I}I− = the maximal element in {J ∈ Ld : J < I}and we call I+ the the upper adjacent of I and I− the lower adjacentof I.199The Refined Filtrations on U(n−w) and KFor an arbitrary I ∈ Ld, we denote byLd≤I = {J ∈ Ld : J ≤ I}UI(n−w) = the subspace of U(n−w) spanned by all YJ such that J ≤ IUnder the linear order on Ld, the {UI(n−w) : I ∈ Ld} form an exhaustivefiltration of U(n−w) by the PBW theorem. And for each I ∈ Ld, let I− be itslower adjacent multi-index as above, thenLd≤I = {I} ∪ Ld≤I− ,and by the PBW theorem, the quotient space UI(n−w)/UI−(n−w) is an onedimensional space spanned by the Y I .We denote byKI = U ⊗ UI(n−w).Obviously the {KI : I ∈ Ld} form an exhaustive filtration of K. Let KI/KI−be the quotient space andKI → KI/KI− (7.22)∑J≤IΦJ · Y J 7→ ΦI · Y ImodKI−be the quotient map. Then the quotient space is linearly isomorphic to U :U ∼−→ KI/KI− (7.23)Φ 7→ Φ · Y ImodKI−The KI is a n∅-Submodule of KThe following lemma is the most crucial result. It tells us: if a finite sumof the form∑J≤I ΦJ · Y J is n∅-torsion, then its “leading term” ΦI · Y I istorsion, and then every term is torsion by iteration.Lemma 7.44. For all I ∈ Ld, the subspace KI of the right U(n∅)-moduleK is stable under the right multiplication of n∅. Hence they are right U(n∅)-submodule of K.Therefore the quotient space KI/KI− is a right U(n∅)-module. The quo-tient map (7.22) is a right U(n∅)-homomorphism, and the isomorphism(7.23) is an isomorphism of right U(n∅)-modules.200Remark 7.45 ((Sketch of the proof)). We prove by induction on thelinear order of Ld. For the initial case, when I = (0, . . . , 0), the KI = U isobviously a n∅-submodule.For an arbitrary I, suppose for all J < I (equivalently J ≤ I−), the KJis a right U(n∅)-submodule (induction hypothesis). To show the KI is n∅-stable, we just need to show the (Φ ·Y I) ·X (i.e. the leading term multipliedby X), is still in KI for all Φ ∈ U , X ∈ n∅.First, by the fortified formula (7.15) in Lemma 7.32, we have(Φ · Y I) ·X = Φ · (Y I ·X)=∑JIΦ · 〈Y J , X〉 · Y I−JWe split the last sum into three parts:∑JI,|J |=0Φ · 〈Y J , X〉 · Y I−J +∑JI,|J |=1Φ · 〈Y J , X〉 · Y I−J+∑JI,|J |>1Φ · 〈Y J , X〉 · Y I−J .1. The first part is a single term∑JI,|J |=0Φ · 〈Y J , X〉 · Y I−J = Φ ·X · Y Isince 〈Y J ,−〉 = id. This “leading term” is still in KI but not in KI− .2. For the third part∑JI,|J |>1 Φ · 〈Y J , X〉 ·Y I−J , we look at each sum-mand. Since |J | ≥ 2, we see |I − J | = |I| − |J | ≤ |I| − 2. Therefore,the third part is in KI− .3. The second term is the hard part to deal with. We just need to showeachΦ · 〈Y J , X〉 · Y I−Jis in KI− for all J I, |J | = 1.Proof. Let I = (i1, . . . , id). Assume J = (0, . . . , 1, . . . , 0) be the multi-indexwith jth component equal to 1 and all the other component equal to 0. ThenY J = Yj201is a single monomial element (the jth element from the basis {Y1, . . . , Yd}of n−w).We need to showΦ · 〈Y J , X〉 · Y I−J = Φ · [Yj , X] · Y I−Jis in the KI− .Without loss of generality, we assume X is a root vector in n∅. Notethat the element [Yj , X] may not be an element in p∅ or n−w (the worstcase: it might be in w−1pw ∩ n∅), hence it may be neither tangent to Gwnor transverse to Gw. The derivative [Yj , X] is regarded as a left invariantvector field [Yj , X]L on G and also on the open subset G≥w. As in 7.1.5, wecan write it as a pointwise “linear combination”[Yj , X]Lx = Dx +d∑i=1Ci(x)Y Li,xwhere Dx is a tangent vector in TxGw and Ci(x) is an algebraic functionof x (and we have seen it only depends on n if we write x = pwn forp ∈ P, n ∈ N+w .)By Lemma 7.40, we have seen the Ci are all zero if i ≥ j, hence theabove linear combination is written as[Yj , X] = D +∑k<jCkYk.Now the original Φ · [Yj , X] · Y I−J is written asΦ · [Yj , X] · Y I−J = Φ ·D · Y I−J +∑k<jΦ · (CkYk) · Y I−J= Φ ·D · Y I−J +∑k<j(Φ · Ck) · Yk · Y I−JThe Ck are algebraic functions, hence Φ · Ck are still distributions in U =S(Gw, V )′. The D is tangent to Gw therefore Φ ·D is also in S(Gw, V )′.By Lemma 7.41, the terms Yk · Y I−J are C-linear combinations of ele-ments of the form Y l11 Yl22 · · ·Y ldd where(l1, . . . , ld) ≤ (i1, . . . , ik−1, ik + 1, ik+1, . . . , ij−1, ij − 1, ij+1, . . . , id)and the latter index is strictly less than I.In sum, the original Φ · [Yj , X] · Y I−J is in KI− .2027.3.7 The Inclusion K[n∅•] ⊂MLetKI[n∅•] = K[n∅•] ∩ KI .Then the {KI[n∅•] : I ∈ Ld} form an exhaustive filtration of K[n∅•]. To showthe inclusion K[n∅•] ⊂M, we just need to showKI[n∅•] ⊂Mfor all I.We show the inclusions KI[n∅•] ⊂ M by induction on the linear orderof I ∈ Ld. First the initial case is obvious: let I = (0, . . . , 0), we seeKI[n∅•] = U ∩ K[n∅•] = U[n∅•] ⊂M.(Induction Hypothesis): For arbitrary I, assume KJ[n∅•] ⊂ M for allJ < I. We show KI[n∅•] ⊂M. Let∑J≤IΦJ · Y J ∈ KI[n∅•]be an arbitrary element, then we need to show all ΦJ are inside U[n∅•].Consider the following commutative diagramKI KI/KI− UKI[n∅•] (KI/KI−)[n∅•] U[n∅•]''By Lemma 7.44, the linear isomorphism U ' KI/KI− is actually an isomor-phism of U(n∅)-modules, hence they have the same torsion submodules:U[n∅•] ' (KI/KI−)[n∅•].The image of∑J≤I ΦJ · Y J in the quotient KI/KI−is in the torsion sub-module (KI/KI−)[n∅•] since the quotient map is U(n∅)-linear by Lemma 7.44,hence ΦI · Y ImodKI− is in the torsion submodule (KI/KI−)[n∅•], then theΦI ∈ U[n∅•].Hence the leading term ΦI · Y I of∑J≤I ΦJ · Y J is in U[n∅•] ⊂ M. Bythe “easy part”: M ⊂ K[n∅•], we see the leading term ΦI · Y I is in KI[n∅•],hence the following sum∑J<IΦJ · Y J =∑J≤IΦJ · Y J − ΦI · Y I203is in the intersection K[n∅•] ∩ KI−= KI−[n∅•]. By the induction hypothesis,each ΦJ for J < I are inside U[n∅•]. In sum, all ΦJ in the sum∑J≤I ΦJ ·Y Jare inside U[n∅•], hence ∑J≤IΦJ · Y J ∈M.204Chapter 8Shapiro’s LemmaSummary of This ChapterIn the last two chapters, we have shown the spaceKer(Resw) = (I≥w/I>w)′is isomorphic to I ′w ⊗ U(n−w), and its n∅-torsion subspace is exactly[(I≥w/I>w)′][n∅•] = (I ′w)[n∅•] ⊗ U(n−w).This chapter is devoted to the study of the torsion subspace (I ′w)[n∅•],or more precisely the annihilators (I ′w)[n∅k]. Our aim is to find the explicitM∅-action on these annihilators.• In 8.1, we use the “annihilator-invariant trick” to show the followingisomorphism (see (8.1))(I ′w)[n∅k] ∼−→ H0(n∅, (Iw ⊗ Fk)′).Then we can study the kth annihilator by studying the zeroth n∅-cohomology on (Iw ⊗ Fk)′.• In 8.2, we show the following isomorphismIw ⊗ Fk ∼−→ SIndN∅N∅∩w−1Pw(σw ⊗ ηk).Then we can replace the (Iw⊗Fk)′ in the 0th cohomology by the dualof the Schwartz induction space SIndN∅N∅∩w−1Pw(σw ⊗ ηk).• In 8.3, we formulate Shapiro’s Lemma and use it to compute the zerothcohomologyH0(N∅, [SIndN∅N∅∩w−1Pw(σw ⊗ ηk)]′).205Combining all three sections, we have the following isomorphisms(I ′w)[n∅k] ' H0(n∅, (Iw ⊗ Fk)′)' H0(N∅, [SIndN∅N∅∩w−1Pw(σw ⊗ ηk)]′)' H0(N∅ ∩ w−1Pw, (Vσw ⊗ Fk)′)Through these isomorphisms, we explicitly express theM∅-action on (I ′w)[n∅k]by the M∅-action on (Vσw ⊗Fk)′. The main result of this chapter is Lemma8.27: the M∅ acts on (I ′w)[n∅k] by ̂σw ⊗ ηk ⊗ γw, where σw is the twistedrepresentation σ ◦Adw, ηk is the conjugation on the finite dimensional quo-tient U(n∅)/(n∅k), the “hat” means the dual representation, and the γw isthe M∅-modular character on the quotient N∅ ∩ w−1Pw\N∅.8.1 Preparation8.1.1 Main Object of This ChapterIn Chapter 6, we have shown the following isomorphism(I≥w/I>w)′ = Ker(Resw) ' I ′w ⊗ U(n−w).This isomorphism is only a linear isomorphism: the U(n−w) is not stableunder P∅-conjugation, hence the right-hand-side cannot be regarded as atensor product of P∅-representation. However we see the n−w is stable underM∅-conjugation, hence the U(n−w) has the M∅-action on it. Then we haveLemma 8.1. The isomorphism Ker(Resw) ' I ′w⊗U(n−w) is M∅-equivariant,when the right-hand-side is endowed with the tensor product M∅-action.Proof. Actually for arbitrary Φ ∈ I ′w, u ∈ U(n−w) and m ∈M∅, we have〈m · (Φ⊗ u), φ〉 = 〈Φ⊗ u,Rm−1φ〉= 〈Φ, RuRm−1φ〉= 〈Φ, Rm−1RmRuRm−1φ〉= 〈Φ, Rm−1RAdm(u)φ〉= 〈m · Φ, RAdm(u)φ〉= 〈(m · Φ)⊗ (Adm(u)), φ〉for all φ ∈ I≥w. (Here RmRuRm−1 = RAdm(u) because the Lie algebra actionis obtained by differentiation of the Lie group action.)206In Chapter 7, we have shown the following equality[(I≥w/I>w)′][n∅•] = (I ′w)[n∅•] ⊗ U(n−w).Since the n∅-torsion subspaces are M∅-stable, this equality is also equivariantunder M∅, when the right-hand-side is endowed with the tensor product M∅-action.The main object of this chapter is to study the M∅-action on the tor-sion subspace (I ′w)[n∅•], or more precisely the M∅-action on the annihilators(I ′w)[n∅k] for all k ∈ Z>0.8.1.2 The Annihilator (I ′w)[n∅k] and the Annihilator-InvariantTrickIn this subsection, we show the following isomorphism by an algebraictrick(I ′w)[n∅k] ' H0(n∅, (Fk ⊗ Iw)′). (8.1)(See below for the meaning of Fk.) Moreover, this isomorphism is P∅-equivariant.The P∅-Representation on U(n∅)/(n∅k)Let (n∅k) be the two-sided ideal of U(n∅) generated by products of kelements in n∅. LetFk = U(n∅)/(n∅k)be the quotient space of U(n∅) modulo the ideal (n∅k).Since the (n∅k) is an ideal, it is a U(n∅)-submodule of U(n∅), and thequotient space Fk has a natural structure of (left) quotient U(n∅)-modules(n∅-modules). More precisely, for a u ∈ U(n∅), let u ∈ Fk be its image underthe quotient map U(n∅)→ Fk. Then the left U(n∅)-action is given byu′ · u := u′u, ∀u, u′ ∈ U(n∅).The quotient module Fk has the following properties.Lemma 8.2. For the quotient spaces Fk, we have• The Fk is a finite dimensional complex vector space.207• The natural conjugation of P∅ on U(n∅) induces a P∅-action on Fk.More precisely, the P∅-action is given byp · u := p · u,where p · u = Adp(u) is the natural conjugation. And this actionmakes Fk into a finite dimensional continuous (hence smooth) P∅-representation. We denote this representation by (ηk, Fk).• The (ηk, Fk) is an algebraic representation of P∅.• The differentiation of the N∅-action on Fk coincides with the U(n∅)-module structure.Remark 8.3. Since Fk is finite dimensional, the topology on it is the canon-ical topology. The strong dual F ′k is exactly the algebraic dual F∗k (also withthe canonical topology since F ∗k is also finite dimensional). All algebraictensor product with Fk or F∗k have a unique topology and they are auto-matically complete.The Annihilator (I ′w)[n∅k]The annihilator (I ′w)[n∅k] is a P∅-stable subspace of the P∅-representationI ′w and a n∅-submodule of I ′w.Given an element Φ ∈ (I ′w)[n∅k], one has a mapLΦ : Fk → I ′wu 7→ u · ΦHere u is the image of an arbitrary u ∈ U(n∅) in the quotient Fk =U(n∅)/(n∅k). The LΦ is a well-defined linear map and it is actually a leftU(n∅)-homomorphism between the two left U(n∅)-modules Fk and I ′w. Thefollowing algebraic facts is easy to prove:Lemma 8.4. The map(I ′w)[n∅k] → HomU(n∅)(Fk, I ′w) (8.2)Φ 7→ LΦis a linear isomorphism.Proof. If LΦ = 0, then u · Φ = 0 for all u ∈ U(n∅). In particular 1 · Φ = 0and Φ = 0, hence (8.2) is injective.Let L ∈ HomU(n∅)(Fk, I ′w), we let Φ = L(1) ∈ I ′w. Obviously (n∅k)·Φ = 0,hence Φ ∈ (I ′w)[n∅k] and it is easy to see L = LΦ, hence (8.2) is surjective.208The Diagonal P∅-Action on HomC(Fk, I ′w)Let HomC(Fk, I′w) be the vector space of linear maps from Fk to I′w. TheFk is a P∅-representation by Lemma 8.2, and the I ′w is the contragredientP∅-representation of the right regular P∅-representation on Iw. We thus havea abstract group action of P∅ on the vector space HomC(Fk, I ′w).More precisely, let L ∈ HomC(Fk, I ′w), we define the P∅-action on it bypL(u) = p · L(p−1u), ∀u ∈ Fk.We call this the diagonal P∅-action on HomC(Fk, I ′w).The subspace HomU(n∅)(Fk, I′w) is stable under the diagonal P∅-action.Actually on both Fk and I′w, the n∅-module structure is obtained by dif-ferentiating the smooth N∅-action. Hence the P∅-action and n∅-action onthe Fk and I′w are compatible in the sense (C-3) in Definition 2.26. Moreprecisely, let L ∈ HomU(n∅)(Fk, I ′w), p ∈ P∅, we havepL(u′u) = p · L[p−1 · u′ · u]= p · L[(p−1 · u) · (p−1 · u)]= p · [(p−1 · u′) · L(p−1 · u)] (L is n∅-linear)= [p · (p−1 · u′)] · [p · L(p−1 · u)]= u′ · [p · L(p−1 · u)]= u′ · pL(u)for all u′ ∈ U(n∅), u ∈ Fk. Hence pL is still n∅-linear.Lemma 8.5. With the HomC(Fk, I′w) and HomU(n∅)(Fk, I′w) endowed withthe above diagonal P∅-actions, we have• The isomorphism (8.2) in Lemma 8.4 is P∅-equivariant.• The natural isomorphismHomC(Fk, I′w) ' F ∗k ⊗ I ′wis P∅-equivariant, with the HomC(Fk, I ′w) endowed with the diagonalP∅-action, and F ∗k ⊗ I ′w endowed with the tensor product P∅-action.• By differentiating the diagonal P∅-action, one has the n∅-action onHomC(Fk, I′w), and its n∅-invariant subspace isH0(n∅,HomC(Fk, I ′w)) = HomU(n∅)(Fk, I′w)and this equality is P∅-equivariant.209• By differentiating the P∅-actions on HomC(Fk, I ′w) and F ∗k ⊗ I ′w, theirn∅-invariant subspaces are isomorphicH0(n∅,HomC(Fk, I ′w)) ' H0(n∅, F ∗k ⊗ I ′w),and this isomorphism is P∅-equivariant.SummaryCombining the above two Lemmas, we have the following sequence ofisomorphisms(I ′w)[n∅k] ' HomU(n∅)(Fk, I ′w) (Lemma 8.4)= H0(n∅,HomC(Fk, I ′w)) (Lemma 8.5)' H0(n∅, F ∗k ⊗ I ′w)' H0(n∅, (Fk ⊗ Iw)′)and all isomorphisms above are P∅-equivariant.Remark 8.6. We can construct a linear map directly:(I ′w)[n∅k] → (Fk ⊗ Iw)′ (8.3)Φ 7→ Φ˜where Φ˜ is given byΦ˜(u⊗ φ) := Φ(Ruφ).And it is easy to see this map is well-defined, P∅-equivariant, with image in-side H0(n∅, (Fk⊗Iw)′), and is exactly the composition of the above sequenceof isomorphisms, hence is an isomorphism.In sum, to study the annihilator (I ′w)[n∅k] and the M∅-action on it, wejust need to compute the 0th cohomology H0(n∅, (Fk ⊗ Iw)′). We will firstshow the Fk ⊗ Iw is isomorphic to a Schwartz induction space, then useShapiro Lemma to compute this 0th cohomology.8.2 The Iw and The Tensor Product Iw ⊗ FkThe main object of this section is to study the tensor product Iw ⊗ Fk,and the M∅-action on it.210• In 8.2.1, we quickly recall the structure of the local Schwartz inductionIw = SIndPwP∅P σ.• In 8.2.2, we define the Schwartz induction SIndN∅N∅∩w−1Pwσw where σwis the twisted representation σ ◦Adw of N∅ ∩w−1Pw on V . We showthere is a natural isomorphism ((8.4) or (8.5)):Iw∼−→ SIndN∅N∅∩w−1Pwσw.• In 8.2.3, we describe the M∅-action on the right-hand-sideSIndN∅N∅∩w−1Pwσwinduced from the M∅-representation Iw by the above isomorphism.• In 8.2.4, we recall the notion of external tensor products of represen-tations, and the following basic property of Schwartz inductions:SIndG1P1 σ1 ⊗̂ SIndG2P2 σ2∼−→ SIndG1×G2P1×P2 σ1 σ2.• In 8.2.5, we apply the above basic property to show the followingisomorphism (8.17):(SIndN∅N∅∩w−1Pwσw)⊗ Fk ∼−→ SIndN∅N∅∩w−1Pw(σw ⊗ ηk|N∅∩w−1Pw).In sum, the Iw ⊗ Fk is isomorphic to the Schwartz inductionSIndN∅N∅∩w−1Pw(σw ⊗ ηk|N∅∩w−1Pw),and we can write down the explicit M∅-action on it.8.2.1 Revisiting the IwRecall that the notation Iw means the local Schwartz induction:Iw = SIndGwP σ,where Gw = PwP∅ is the double coset corresponding to the representativew ∈ [WΘ\W ].211Structure of the Double Coset Gw = PwP∅By abuse of notation, we denote the fixed representative of w ∈ [WΘ\W ]in G (actually in NG(S)) by the same w.We regard Gw as an abstract real variety. For every p ∈ P, q ∈ P∅, wedenote the point pwq ∈ PwP∅ byxpwq ∈ Gw.Remark 8.7. The reader can treat it as the point pwq ∈ G, but we wantto emphasize that the expression pwq is not unique.We have the following easy facts:PwP∅ = PwN∅= Pw(N∅ ∩ w−1Pw) · (N∅ ∩ w−1NPw)= Pw(N∅ ∩ w−1NPw)= PwN+w(Recall that N+w is exactly N∅ ∩ w−1NPw as defined in Chapter 6.)Lemma 8.8. The last expression in the above align is unique. More pre-cisely, the mapP ×N+w → Gw = PwP∅(p, n) 7→ pwnis an isomorphism of real affine varieties and smooth manifolds. In partic-ular, every element in Gw is uniquely written as xpwn for some p ∈ P andn ∈ N+w .Functions in IwThe Iw = SIndGwP σ is defined to be the image space of the followingintegration mapS(Gw, V )→ C∞(Gw, V, σ), f 7→ fσ,where C∞(Gw, V, σ) = {f ∈ C∞(Gw, V ) : f(px) = σ(p)f(x),∀p ∈ P, x ∈Gw} and fσ(x) :=∫P σ(p−1)f(px)dp is the σ-mean value function.212The topology on SIndGwP σ is the quotient topology, and in particular it isa nuclear Fre´chet space, and one has the following surjective homomorphismof TVS and P∅-representations:S(Gw, V ) SIndGwP σ.By the above definition, the Iw = SIndGwP σ is contained in C∞(Gw, V, σ).In particular, a function φ in Iw is a smooth function and satisfies the “σ-rule”:φ(px) = σ(p)φ(x), ∀p ∈ P, x ∈ Gw.8.2.2 The Iw is Isomorphic to SIndN∅N∅∩w−1PwσwWe denote by (σw, V ) the following representation of N∅∩w−1Pw on V :σw(u)v := σ(wuw−1)v, ∀v ∈ V, u ∈ N∅ ∩ w−1Pw.(Note that σ ◦ Adw is a twisted representation of w−1Pw on V , the aboveσw is its restriction to the subgroup N∅ ∩ w−1Pw.)The representation σw is actually a (real) algebraic representation ofN∅ ∩ w−1Pw on V . In particular, it is of moderate growth, and one candefine as in Chapter 4 the Schwartz inductionSIndN∅N∅∩w−1Pwσw,which is the image of S(N∅, V ) under the σw-mean value map:S(N∅, V )→ SIndN∅N∅∩w−1Pwσwf 7→ fσwwhere fσwis given byfσw(n) =∫N∅∩w−1Pwσw(u−1)f(un)du,∀n ∈ N∅.Similar to the Iw, a function ψ in SIndN∅N∅∩w−1Pwσw is a smooth functionon N∅, satisfying the σw-rule:ψ(un) = σw(u)ψ(n), ∀u ∈ N∅ ∩ w−1Pw, n ∈ N∅.213The Map from Iw to SIndN∅N∅∩w−1PwσwWe can define the following mapIw = SIndGwP σ → C∞(N∅, V, σw) (8.4)φ 7→ φ̂where the φ̂ is given byφ̂(n) = φ(xwn),∀n ∈ N∅.The φ̂ is obviously a smooth V -valued function on N∅, and satisfying theσw-rule: for all u ∈ N∅ ∩ w−1Pw, n ∈ N∅, one hasφ̂(un) = φ(xwun)= φ(xwuw−1wn)= σ(wuw−1)φ(xwn) (φ satisfies the σ-rule)= σw(u)φ̂(n)The Image of (8.4)We showLemma 8.9. The image of the map (8.4) is in the Schwartz induction spaceSIndN∅N∅∩w−1Pwσw. Therefore, we have an isomorphism (of TVS):Iw∼−→ SIndN∅N∅∩w−1Pwσw (8.5)φ 7→ φ̂First we have the following diagramC∞(N∅, V, σw)Iw SIndN∅N∅∩w−1PwσwS(N+w , V )(8.4)(4.37)'?(4.37)'214In this diagram, the upper-right arrow is the inclusion, and the two down-slash arrows are isomorphisms given by the Lemma 4.92, since both Gw andN∅ are isomorphic (as real algebraic varieties) to direct products:Gw ' P ×N+wN∅ ' (N∅ ∩ w−1Pw)×N+wTo show the image of (8.4) is in SIndN∅N∅∩w−1Pwσw, we just need to showthe composition of the isomorphism Iw → S(N+w , V ) and the inverse iso-morphism S(N+w , V )→ SIndN∅N∅∩w−1Pwσw is exactly the map (8.4).Proof. The map Iw → S(N+w , V ) is given by (4.37) in Lemma 4.92. Pick aφ ∈ Iw, its image in S(N+w , V ) is exactlyφ(n) := φ(xwn) = φ(wn).The proof of Lemma 4.92 gives us an explicit inverse of the isomorphismSIndN∅N∅∩w−1Pwσw → S(N+w , V ). Namely, let φ ∈ S(N+w , V ) be an arbi-trary Schwartz function on N+w , let γ ∈ C∞c (N∅ ∩ w−1Pw) be an arbitraryC-valued smooth function with compact support such that γ(e) = 1 and∫N∅∩w−1Pw γ(u)du = 1. Then one can first define a function Φ ∈ S(N∅, V )byΦ(un) = γ(u)σw(u)φ(n), ∀u ∈ N∅ ∩ w−1Pw, n ∈ N+w .(Here note N∅ = (N∅ ∩w−1Pw) ·N+w is a direct product of manifolds.) It iseasy to see the Φ is Schwartz on N∅. Now we consider the σw-mean valuefunction of Φ:Φσw(z) =∫N∅∩w−1Pwσw(a−1)Φ(az)da, ∀z ∈ N∅It is easy to check the Φσw ∈ SIndN∅N∅∩w−1Pwσw is independent of thechoice of γ as in the proof of Lemma 4.92. Actually for all u ∈ N∅ ∩215w−1Pw, n ∈ N+w , one hasΦσw(un) =∫N∅∩w−1Pwσw(a−1)Φ(aun)da=∫N∅∩w−1Pwσw(a−1)[γ(au)σw(au)φ(n)]da=∫N∅∩w−1Pwγ(au)σw(u)φ(n)da= [∫N∅∩w−1Pwγ(au)da]σw(u)φ(n)= σw(u)φ(n)Obviously Φσw |N+w = φ, hence Φσwis indeed the image under the inverseisomorphism S(N+w , V )→ SIndN∅N∅∩w−1Pwσw.Now the composition of the two maps Iw → S(N+w , V ) and S(N+w , V )→SIndN∅N∅∩w−1Pwσw sends the φ to Φσw. It is easy to seeΦσw= φ̂ on N∅Therefore the composition of the two isomorphisms Iw → S(N+w , V ) andS(N+w , V )→ SIndN∅N∅∩w−1Pwσw is exactly the map (8.4).8.2.3 The Isomorphism (8.5) is M∅-EquivariantNote that the local Schwartz induction Iw = SIndGwP σ is a representationof P∅ under the right regular P∅-action, denoted byRpφ, ∀φ ∈ Iw, p ∈ P∅.The isomorphism (8.5) transport the right regular P∅-action on Iw to aP∅-action on SIndN∅N∅∩w−1Pwσw. We can explicitly write down the P∅-actionon SIndN∅N∅∩w−1Pwσw.The N∅-Action on SIndN∅N∅∩w−1PwσwThe SIndN∅N∅∩w−1Pwσw is obviously aN∅-representation with right regularN∅-action. The following lemma is trivial.Lemma 8.10. With both sides endowed with the right regular N∅-actions,the isomorphism (8.5) is N∅-equivariant.216The M∅-Action on SIndN∅N∅∩w−1PwσwWe compute the M∅-action on SIndN∅N∅∩w−1Pwσw. Let φ ∈ Iw be arbi-trary, and let φ̂ ∈ SIndN∅N∅∩w−1Pwσw be its image under the isomorphism(8.5). For arbitrary m ∈M∅, n ∈ N∅, we haveR̂mφ(n) = (Rmφ)(xwn)= φ(xwnm)= φ(xwmw−1wm−1nm)= σ(wmw−1)φ(xwm−1nm)= σw(m)φ̂(m−1nm)In sum, the M∅-action on SIndN∅N∅∩w−1Pwσw is given bym · ψ = σw(m) ◦ ψ ◦Adm−1 (8.6)for all ψ ∈ SIndN∅N∅∩w−1Pwσw,m ∈ M∅. Actually this M∅-action is well-defined on the entire C∞(N∅, V, σw), and we have just shown the subspaceSIndN∅N∅∩w−1Pwσw is stable under this M∅-action.Lemma 8.11. With the right regular M∅-action on Iw and the M∅-action onSIndN∅N∅∩w−1Pwσw defined in (8.6), the isomorphism (8.5) is M∅-equivariant.8.2.4 External Tensor ProductsFor i = 1, 2, letGi = a real algebraic groupsPi = an algebraic subgroup of Gi(σi, Vi) = a smooth nuclear Fre´ceht representations of PiExternal Tensor Products of RepresentationsDefinition 8.12. The following representation of P1 × P2 on V1 ⊗̂ V2 iscalled the external tensor product of σ1 and σ2, and is denoted by(σ1 σ2, V1 ⊗̂ V2):σ1 σ2(p1, p2)(v1 ⊗ v2) := [σ1(p1)v1]⊗ [σ2(p2)v2],for all pi ∈ Pi, vi ∈ Vi, i = 1, 2.217Remark 8.13. When P1 = P2, the restriction of σ1 σ2 to the diagonalsubgroup of P1 × P2 is exactly the tensor product σ1 ⊗ σ2.It is easy to seeLemma 8.14. The representation σ1 σ2 is a smooth NF-representation.Moreover, if σ1, σ2 are of moderate growth, then so is σ1 σ2.Tensor Product of Schwartz InductionsSuppose σ1, σ2 are of moderate growth, then one has three Schwartzinduction spaces:SIndG1P1 σ1, SIndG2P2 σ2, SIndG1×G2P1×P2 σ1 σ2.For i = 1, 2, let φi ∈ SIndGiPi σi be two arbitrary functions, then one candefine a smooth function on G1 ×G2 with values in V1 ⊗̂ V2:φ1 φ2(g1, g2) := φ1(g1)⊗ φ2(g2), ∀(g1, g2) ∈ G1 ×G2. (8.7)Then we haveLemma 8.15. The function φ1 φ2 is in SIndG1×G2P1×P2 σ1 σ2, and the mapφ1 ⊗ φ2 7→ φ1 φ2 extends to an isomorphism of TVS:SIndG1P1 σ1 ⊗̂ SIndG2P2 σ2 → SIndG1×G2P1×P2 σ1 σ2 (8.8)φ1 ⊗ φ2 7→ φ1 φ28.2.5 Tensor Product of Schwartz InductionsIn this subsection, we show a Lemma about tensor product of Schwartzinduction. We will apply this Lemma to the tensor product Iw ⊗ Fk. Theanalogue of this Lemma in the ordinary induction picture is well-known.Let G,P be the same as in 4.6, and let(σ, V ) = a nuclear Fre´chet representation of P of moderate growth(η, F ) = a finite dimensional algebraic representation of G(η|P , F ) = the restriction of η to the subgroup PSIndGPσ = the Schwartz induction space of σIn particular, the η and η|P are of moderate growth.218Let φ ∈ SIndGPσ, v ∈ F , we define the following function in C∞(G,V ⊗F ):φv(g) := φ(g)⊗ η(g)v, ∀g ∈ G. (8.9)The main result of this subsection is:Lemma 8.16. The function φv is in SIndGP (σ ⊗ η|P ). And we have thefollowing isomorphism of TVS:SIndGPσ ⊗ F → SIndGP (σ ⊗ η|P ) (8.10)φ⊗ v 7→ φvThis map is an isomorphism of G-representations, where the left hand sideis the tensor product representation and right hand side is the right regularrepresentation.We first show the function φv is indeed in SIndGP (σ⊗ η|P ), by finding itspreimage in the Schwartz function space S(G,V ⊗ F ). Then we show theabove map is an isomorphism.Tensor Product on Schwartz Function SpacesFor a f ∈ S(G,V ) and a v ∈ F , we can define the following smoothfunction on G with values in V ⊗ Ffv(g) := f(g)⊗ η(g)v, ∀g ∈ G. (8.11)We claim:Lemma 8.17. The function fv is in S(G,V ⊗ F ).To prove this Lemma, we just need to apply the following Lemma to(ρ, U) = (1VG ⊗ η, V ⊗ F ) where 1VG is the trivial representation of G on V .Lemma 8.18. Let G be a real point group of a linear algebraic group, and(ρ, U) be a representation of G of moderate growth. For an arbitrary f ∈S(G,U), we define the functionρf(g) := ρ(g)f(g), ∀g ∈ G. (8.12)Then ρf is in S(G,U).Proof. One can easily check, for all u ∈ U(g), the[Ru(ρf)](g) = ρ(g)[Ruf(g)].Since ρ is of moderate growth, the Ruρf is bounded on entire G. SinceU(g) generates the entire ring of Nash differential operators (and algebraicdifferential operators), we see ρf is rapidly decreasing on G.219The φv is in SIndGP (σ ⊗ η|P )We show the first part of Lemma 8.16: for all φ ∈ SIndGPσ, and v ∈ F ,the φv defined by (8.9) is in the Schwartz induction space SIndGP (σ ⊗ η|P ).By definition of the Schwartz induction, one can find a Schwartz functionf ∈ S(G,V ) such that fσ = φ. Then we consider the function fv ∈ S(G,V ⊗F ) defined in (8.11), and its σ ⊗ η-mean value function (fv)σ⊗η, and we see(fv)σ⊗η(g) =∫P(σ ⊗ η)(p−1)fv(pg)dp=∫P(σ ⊗ η)(p−1)[f(pg)⊗ η(pg)v]dp=∫P[σ(p−1)f(pg)]⊗ [η(p−1)η(pg)v]dp=∫P[σ(p−1)f(pg)]⊗ [η(g)v]dp= [∫Pσ(p−1)f(pg)dp]⊗ [η(g)]= fσ(g)⊗ η(g)v= φ(g)⊗ η(g)v= φv(g)Therefore, the φv is the image of fv under the mean value map S(G,V ⊗F )→ SIndGP (σ ⊗ η|P ).Relation With External Tensor ProductFor a ψ ∈ SIndGP (σ⊗ η|P ), one can define the following smooth functionon G×G with values in V ⊗ F :φ∧(g1, g2) := [id⊗ η(g2g−11 )]ψ(g1), ∀(g1, g2) ∈ G×G. (8.13)Also for a Ψ ∈ SIndG×GP×Gσ η, one can define the following smoothfunction on G with values in V ⊗ F :Ψ∨(g) := Ψ(g, g), ∀g ∈ G. (8.14)Then we haveLemma 8.19. With the above notations,(1) The ψ∧ is in SIndG×GP×Gσ η.220(2) The Ψ∨ is in SIndGP (σ ⊗ η|P ).(3) The two maps ψ 7→ ψ∧ and Ψ 7→ Ψ∨ are mutually inverse, therefore wehave the following isomorphism of TVS:SIndGP (σ ⊗ η|P ) ∼−→ SIndG×GP×Gσ η (8.15)(4) The isomorphism (8.15) is G-equivariant, when the left-hand-side hasthe right regular G-action, and right-hand-side has the diagonal rightregular G-action.The Map (8.10) is an IsomorphismWe show the (8.10) is an isomorphism.First we note F ' SIndGGη by Lemma 4.93, and this isomorphism com-bined with the Lemma 8.15, gives the following isomorphismSIndGPσ ⊗ F '−→ SIndG×GP×Gσ η (8.16)φ⊗ v 7→ φ vwhere the function φ v is given byφ v(g1, g2) := φ(g1)⊗ η(g2)v, ∀(g1, g2) ∈ G×G.We then have the following diagramSIndGPσ ⊗ F SIndG×GP×Gσ ηSIndGP (σ ⊗ η|P )(8.16)(8.10) (8.15)And we have seen the (8.15) and (8.16) are isomorphisms, we just need toverify the above diagram commutates, i.e. (φv)∧ = φ v. Actually,(φv)∧(g1, g2) = [id⊗ η(g2g−11 )]φv(g1)= [id⊗ η(g2g−11 )]φ(g1)⊗ η(g1)v= φ(g1)⊗ η(g2)v= φ v(g1, g2)221Application to Iw ⊗ FkApplying the Lemma 8.16, toG = N∅P = N∅ ∩ w−1Pw(σ, V ) = (σw, V )(η, F ) = (ηk, Fk)where Fk = U(n∅)/(n∅k), we have the following LemmaLemma 8.20. For any φ ∈ SIndN∅N∅∩w−1Pwσw and v ∈ Fk, the functionφv(n) := φ(n)⊗ ηk(n)v, ∀n ∈ N∅,is in SIndN∅N∅∩w−1Pw(σw ⊗ ηk). The map(SIndN∅N∅∩w−1Pwσw)⊗ Fk → SIndN∅N∅∩w−1Pw(σw ⊗ ηk) (8.17)φ⊗ v 7→ φvis an isomorphism of TVS.By abuse of notation, we write ηk instead of ηk|N∅∩w−1Pw, to keep allequations short. This will not cause any ambiguity.The M∅-Action on SIndN∅N∅∩w−1Pw(σw ⊗ ηk)Combining the isomorphisms (8.5) in Lemma 8.9 and (8.17) in Lemma8.20, we have the following isomorphismIw ⊗ Fk ∼−→ SIndN∅N∅∩w−1Pw(σw ⊗ ηk) (8.18)φ⊗ v 7→ φ̂vwhere the φ̂ is defined by φ̂(n) = φ(xwn), and the φ̂v(n) is defined by φ̂v(n) =φ̂(n)⊗ ηk(n)v.The left-hand-side Iw ⊗Fk has the tensor product M∅-action, where theM∅ acts on Iw by right regular action, and on Fk by the conjugation whichis also denoted by ηk.222Now we want to describe the corresponding M∅-action on the right-hand-side SIndN∅N∅∩w−1Pw(σw ⊗ ηk). Obviously, it is given bym · φ̂v = (R̂mφ)ηk(m)v= R̂mφ(n)⊗ ηk(n)ηk(m)v= Rmφ(xwn)⊗ ηk(n)ηk(m)vWe should write the (R̂mφ)ηk(m)v in a more explicit form. The firstcomponent is given by (8.6):R̂mφ(n) = σw(m)φ̂(Adm−1n).The second component is given byηk(n)ηk(m)v = ηk(m)ηk(Adm−1n)v.Therefore, we havem · φ̂v = (σw ⊗ ηk)(m) · φ̂v(Adm−1n).And in general, we haveLemma 8.21. The M∅-action on the SIndN∅N∅∩w−1Pw(σw ⊗ ηk) induced bythe isomorphism (8.18) is given bym · Φ = (σw ⊗ ηk)(m) ◦ Φ ◦Adm−1, (8.19)for all Φ ∈ SIndN∅N∅∩w−1Pw(σw ⊗ ηk) and m ∈M∅.8.3 Shapiro’s LemmaIn the last two sections, we have shown the isomorphism(I ′w)[n∅k] ' H0(n∅, (Iw ⊗ Fk)′)and the following isomorphism between Iw ⊗ Fk and a Schwartz inductionspaceIw ⊗ Fk ' SIndN∅N∅∩w−1Pw(σw ⊗ ηk).Therefore, the computation of the kth annihilator (I ′w)[n∅k], is reduced tothe computation of the space of n∅-invariant distributions on the Schwartzinduction SIndN∅N∅∩w−1Pw(σw ⊗ ηk), namely the following spaceH0(n∅, [SIndN∅N∅∩w−1Pw(σw ⊗ ηk)]′).223• In 8.3.1, we formulate a version of Shapiro’s lemma (Theorem 8.23).Let G,H be unimodular real algebraic groups, and (ρ, U) be a smoothnuclear representation of H of moderate growth. Let (ρ̂, U ′) be thedual representation of (ρ, U), and SIndGHρ be the Schwartz induction.Then the natural map (see (8.21))H0(H, ρ̂)→ H0(G, (SIndGHρ)′)is an isomorphism.• In 8.3.2, we apply Shapiro’s lemma to the Schwartz inductionSIndN∅N∅∩w−1Pw(σw ⊗ ηk).Since the group N∅ and its subgroup N∅∩w−1Pw are unimodular andcohomological trivial, we can apply the Shapiro’s Lemma which tellsus the H0(n∅, (Iw ⊗ Fk)′) is exactly the following spaceH0(n∅ ∩ w−1pw, (Vσw ⊗ Fk)′).8.3.1 Shapiro’s LemmaIn this subsection, we state the version of Shapiro’s Lemma which wewill apply. To simplify the understanding of Shapiro Lemma, we state aweaker version of it, and the general version could be reduced to this weakerversion. This means we put some strong conditions on the groups, and inthis subsection, we temporarily stick to the following notations:G = a real algebraic group which is unimodularH = a (unimodular) closed algebraic subgroup of G(ρ, U) = smooth nuclear Fre´chet representation of Hwhich is of moderate growthS(G,U) = the space of Schwartz V -valued functions on GSIndGHρ = the Schwartz induction spaceSince G,H are both unimodular, one has a right G-invariant measure onthe quotient H\G, denoted by dx. Further more, we assume the H has acomplement in G, i.e. G has a real subvariety Q such that the multiplicationmap H ×Q→ G is an isomorphism of real varieties and manifolds.Let (ρ̂, U ′) be the contragredient H-representation on the strong dualU ′, and let H0(H,U ′) be the space of H-invariant vectors in U ′. Given a224vector λ ∈ H0(H,U ′) and a φ ∈ SIndGHρ, we consider the following functionon G:〈λ, φ(−)〉 : g 7→ 〈λ, φ(g)〉, ∀g ∈ G,where 〈, 〉 : U ′ ×U → C is the pairing between U ′ and U . This function hasthe following properties:Lemma 8.22. The 〈λ, φ(−)〉 is constant on each H-coset, hence it factorsthrough a smooth function on the quotient manifold H\G, which is denotedby〈λ, φ〉 : xg 7→ 〈λ, φ(g)〉, ∀xg ∈ H\G.Here xg means the point on H\G represented by the group element g ∈ G.Moreover, the function 〈λ, φ〉 is integrable on H\G, and the map:Υλ : SIndGHρ→ C (8.20)φ 7→∫H\G〈λ, φ〉(xg)dxgis a continuous linear functional on SIndGHρ, which is G-invariant.Proof. The 〈λ, φ(g)〉 is constant on each double coset because λ is invariantunder H. The 〈λ, φ〉 is integrable because it is a Schwartz function onH\G. The functional Υλ is G-invariant because the measure dx is rightG-invariant.Therefore, we have a linear mapH0(H,U ′)→ H0(G, (SIndGHρ)′) (8.21)λ 7→ Υλand it is an isomorphism by the following version of Shapiro’s LemmaTheorem 8.23 (Casselman). The map (8.21) is an isomorphism.The proof is unpublished now, and we omit it.Remark 8.24. The traditional Shapiro’s Lemma says the above isomor-phism also holds for higher cohomology. We will only apply the special caseof zeroth cohomology, which is actually a variation of Frobenius reciprocity.Remark 8.25. Our original work applied the version of Shapiro’s Lemmain [2] (p182 Theorem 4.0.13). But the version there requires the represen-tation σ to be a Nash representation, in particular the σ need to be finitedimensional, which cannot be generalized to infinite dimensional σ. Thisis a strong confinement of our work, therefore we abandon the old proofand switch to the current version of Shapiro’s Lemma which also applies toinfinite dimensional σ.2258.3.2 Applying Shapiro’s LemmaWe need to study the M∅-structure on the annihilators (I ′w)[n∅k], and wehave shown it is linearly isomorphic toH0(n∅, (Iw ⊗ Fk)′) ' H0(n∅, [SIndN∅N∅∩w−1Pw(σw ⊗ ηk)]′).Since the N∅ and its subgroup N∅ ∩w−1Pw are cohomologically trivial,(i.e. they are isomorphic to Euclidean spaces by the exponential maps), wehave the isomorphisms of functorsH0(N∅,−) = H0(n∅,−)H0(N∅ ∩ w−1Pw,−) = H0(n∅ ∩ w−1pw,−)In particular, we haveH0(n∅, [SIndN∅N∅∩w−1Pw(σw ⊗ ηk)]′) = H0(N∅, [SIndN∅N∅∩w−1Pw(σw ⊗ ηk)]′)Combining this with Shapiro’s Lemma (Theorem 8.23):H0(N∅, [SIndN∅N∅∩w−1Pw(σw ⊗ ηk)]′) ' H0(N∅ ∩ w−1Pw, (Vσw ⊗ Fk)′),(Here the Vσw means the V with the group action through the σw) we have(I ′w)[n∅k] ' H0(n∅, (Iw ⊗ Fk)′)' H0(n∅, [SIndN∅N∅∩w−1Pw(σw ⊗ ηk)]′)= H0(N∅, [SIndN∅N∅∩w−1Pw(σw ⊗ ηk)]′)' H0(N∅ ∩ w−1Pw, (Vσw ⊗ Fk)′)In sum, we can describe the elements in the annihilators (I ′w)[n∅k] by thecohomologiesH0(N∅ ∩ w−1Pw, (Vσw ⊗ Fk)′).And the above sequence of isomorphisms also transplant the natural M∅-action on (I ′w)[n∅k] onto H0(N∅ ∩ w−1Pw, (Vσw ⊗ Fk)′). The last step isto describe the M∅-action on H0(N∅ ∩ w−1Pw, (Vσw ⊗ Fk)′) in terms of(Vσw ⊗ Fk)′.Remark 8.26. The (Vσw ⊗ Fk)′ is the dual of Vσw ⊗ Fk, which is a tensorproduct of two M∅-representations: the M∅ acts on Vσw through σw (sinceM∅ ⊂ w−1Pw), and on the Fk through ηk. However we will see the M∅-action obtained from (I ′w)[n∅k] through the above isomorphisms, is NOT thedual of σw ⊗ ηk.226The M∅-Action From (I ′w)[n∅k]We have the shown two isomorphisms to H0(N∅, [SIndN∅N∅∩w−1Pw(σw ⊗ηk)]′):(I ′w)[n∅k] ∼−→ H0(N∅, [SIndN∅N∅∩w−1Pw(σw ⊗ ηk)]′)H0(N∅ ∩ w−1Pw, (Vσw ⊗ Fk)′) ∼−→ H0(N∅, [SIndN∅N∅∩w−1Pw(σw ⊗ ηk)]′)Therefore the M∅-structure on (I ′w)[n∅k] could be described by the M∅-structure on (Vσw ⊗ Fk)′. Unfortunately, the inverse of the isomorphism(8.21) is not explicit. Thus to compare the M∅-actions on (I ′w)[n∅k] andH0(N∅ ∩ w−1Pw, (Vσw ⊗ Fk)′), we have to send these two M∅-actions toH0(N∅, [SIndN∅N∅∩w−1Pw(σw ⊗ ηk)]′) and compare on this space.First we look at the M∅-action from (Vσw ⊗ Fk)′. Let λ ∈ H0(N∅ ∩w−1Pw, (Vσw ⊗ Fk)′) ⊂ (Vσw ⊗ Fk)′, and m ∈M∅ be arbitrary element. LetΥm·λ ∈ H0(N∅, [SIndN∅N∅∩w−1Pw(σw ⊗ ηk)]′) be the integration distributionintroduced in Lemma 8.22, then for all Φ ∈ SIndN∅N∅∩w−1Pw(σw ⊗ ηk), wehave〈Υm·λ,Φ〉 =∫N∅∩w−1Pw\N∅〈m · λ,Φ(n)〉dxn=∫N∅∩w−1Pw\N∅〈λ, [(σw ⊗ ηk)(m−1)] · Φ(n)〉dxnSecond we look at the M∅-action from (I ′w)[n∅k], we have (λ,m,Υλ,Φ asabove)〈m ·Υλ,Φ〉 = 〈Υ,m−1 · Φ〉=∫N∅∩w−1Pw\N∅〈λ, [(σw ⊗ ηk)(m−1)] · Φ(mnm−1)〉dxnLet n′ = mnm−1, then we have〈m ·Υλ,Φ〉 =∫N∅∩w−1Pw\N∅〈λ, [(σw ⊗ ηk)(m−1)] · Φ(n′)〉dxm−1n′m= γw(m)∫N∅∩w−1Pw\N∅〈λ, [(σw ⊗ ηk)(m−1)] · Φ(n′)〉dxn′= γw(m)〈Υm·λ,Φ〉227where the γw is a character of M∅ given byγw(m) = det(Adn∅∩w−1pw\n∅(m−1)), ∀m ∈M∅. (8.22)In particular, the γw restricted to the A∅ is given by the following char-acter of A∅:δw(a) =∏α∈Σ+α∈w−1(Σ−−Σ−Θ)α(a)mα , ∀a ∈ A∅, (8.23)where mα is the multiplicity of α (Note that the roots are restricted roots,with multiplicities).Now we can see the difference between two M∅-actions on H0(N∅ ∩w−1Pw, (Vσw ⊗ Fk)′): the M∅-action on H0(N∅, [SIndN∅N∅∩w−1Pw(σw ⊗ ηk)]′)is expressed by the M∅-action on (Vσw ⊗ Fk)′ twisted by γw.Lemma 8.27. The M∅ acts on (I ′w)[n∅k], through ̂σw ⊗ ηk ⊗ γw, where γwis the character in (8.22).228Chapter 9Application: FiniteDimensional (σ, V )Summary of This ChapterIn this chapter, we combine tools developed in previous chapters, toreproduce irreducibility theorems (Theorem 7.2a on page 193 and Theorem7.4 on page 203) in [15].Setting and notationsFor groups, we keep the notations as previous chapters. In particular,for the parabolic subgroup P = MPNP , we also write it as PΘ = MΘNΘwhen we need to emphasize the subset Θ. In this chapter, let(τ, V ) = a irreducible unitary representation of MPand we extend it trivially to the entire P . The τ is obviously a Harish-Chandra representation, in particular the V is nuclear.Letσ = τ ⊗ δ1/2Pbe the representation of P on V obtained by twisting the τ by δ1/2P . Then thenormalized unitary (Hilbert) induction of IndGP τ is infinitesimally equivalentto:I = C∞IndGPσ = SIndGPσ.We will apply the results in Chapter 5, 6, 7 and 8 to the σ = τ ⊗ δ1/2P .9.1 Degenerate Principal SeriesIn this section, we prove an analogue results (Theorem 9.6) of Bruhat’sTheorem 7.4 in [15], on the irreducibilities of degenerate principal series.229• In 9.1.1, we show the restriction of τ to the A∅ is a direct sum ofA∅-characters that are WΘ-conjugate to each other.• In 9.1.2, we formulate our analogous version of Bruhat’s Theorem 7.4,and give an outline of the proof.• In 9.1.3, we complete the first step of the proof, by showingHomP (V′, (I/I>e)′) = C.The Theorem 9.8 also tells us the interesting phenomenon occurs onnon-identity double cosets when the I is reducible.The second step to prove the Theorem 9.6 is completed in the next section.9.1.1 A∅-Spectrum on (τ, V )In this subsection, let (τ, V ) be a finite dimensional irreducible unitaryrepresentation of MP . We follow the [15] to describe the A∅-spectrum onthe representation (τ, V ).The main result in this subsection is Lemma 9.2, which says the restric-tion of τ to the A∅ splits into a direct sum of unitary characters of A∅,moreover these A∅-characters are all WΘ-conjugate to each other.For the parabolic subgroup P = PΘ = MΘNΘ, let PΘ =◦MΘAΘNΘ beits Langlands decomposition, i.e. NΘ is the unipotent radical, AΘ is thesplit component, and ◦MΘ is the intersection of kernels of real characters onMΘ. The Levi component is the direct product of◦MΘ and AΘ:MΘ =◦MΘ ×AΘ,and AΘ is exactly the center of MΘ.For the irreducible unitary representation (τ, V ), we know it is trivial onNΘ. Let◦τ = the restriction of τ to ◦MΘχc = the restriction of τ to AΘThen the ◦τ is an irreducible unitary representation of ◦MΘ on V , the χc isa unitary character of AΘ (called the restricted character of τ), and theτ is written asτ = ◦τ ⊗ χc ⊗ 1. (9.1)230The Restriction of τ to ◦MΘ0Let ◦MΘ0 be the identity component of ◦MΘ, and let◦τ0 = the restriction of τ to ◦MΘ0.In general, the ◦τ0 is a unitary representation of ◦MΘ0, but it may not beirreducible.Since the ◦τ0 is a finite dimensional unitary representation of ◦MΘ0,it decomposes into a direct sum of finite dimensional irreducible unitaryrepresentations of ◦MΘ0. The ◦MΘ0-irreducible constituents on V are ◦MΘ-conjugate to each other.Actually, we just need to apply the following Lemma to G = ◦MΘ, H =◦MΘ0:Lemma 9.1. Let G be a Lie group, H be a normal subgroup of G with finiteindex. Let (pi, V ) be a finite dimensional irreducible unitary representation ofG. Then the restriction (pi|H , V ) is a finite direct sum of irreducible unitaryrepresentations of H, and the H-irreducible constituents are G-conjugate toeach other.Proof. Since V is finite dimensional, one can find a minimal nonzero H-invariant subspace of V , denoted by V0. Then V0 is obviously a irreduciblerepresentation of H. LetG(V0) = {g ∈ G : pi(g)V0 ⊂ V0}.Then the G(V0) is a subgroup of G containing H, and H is also normal inG(V0).Let {g0 = e, g1, . . . , gl} be a set of representatives of the right cosetsin G/G(V0). Then obviously each pi(gi)V0 is a H-invariant subspace of V(of the same dimension as V0) since H is normal, and they are irreduciblerepresentations of H.It is easy to see: for i 6= j, either pi(gi)V0 ∩ pi(gj)V0 = {0} or pi(gi)V0 =pi(gj)V0. The latter cannot happen, otherwise g−1i gj ∈ Gpi contradicting toi 6= j. Hence one has a direct suml⊕i=0pi(gi)V0,which is G-invariant subspace of V , hence equals to the entire V since V isG-irreducible.231By applying the above Lemma, we may denote the irreducible con-stituents of (◦τ |◦MΘ0 , V ) by (ρ0, V0), . . . , (ρl, Vl):◦τ0 = ◦τ |◦MΘ0 =l⊕i=0(ρi, Vi).As above, let ◦MΘ(V0) be the subgroup of ◦MΘ fixing the subspace V0. Andfor each i 6= 0, there is an element gi ∈ ◦MΘ representing its ◦MΘ(V0)-cosetin ◦MΘ, such that ρi = ρ0 ◦Adgi.The A∅-Spectrum on (τ, V )As in [15], the restriction of the ◦MΘ0-irreducible constituent (ρ0, V0)to the subgroup ◦MΘ ∩ A∅ is a unitary character of ◦MΘ ∩ A∅, denoted byχ0. Similarly, for each i = 1, . . . , l, the restriction of ρi to◦MΘ ∩ A∅ is aunitary character χi, we thus have l + 1-unitary characters of◦MΘ ∩ A∅:χ0, χ1, . . . , χl.We have seen the ρi are all◦MΘ-conjugate to each other, i.e. for eachi = 1, . . . , l there is a gi ∈ ◦MΘ such that ρi = ρ0 ◦Adgi. Therefore, for thecharacters χi, one also hasχi = χ0 ◦Adgi, i = 0, 1, . . . , l.Moreover, one can choose the gi from the normalizer NG(A∅), and letwi be its image in the Weyl group W . One can see the wi is actually inWΘ since it centralizes the AΘ. Therefore, we see the characters χi are allWΘ-conjugate to each other:χi = χ0 ◦Adwi, i = 0, . . . , l.Since the A∅ is the direct product of AΘ and ◦MΘ ∩A∅:A∅ = (◦MΘ ∩A∅)×AΘ,each χi combined with χc gives a character of A∅. We denote it byχi = χi · χc.It is easy to see χi is exactly the wi-conjugation of χ0:χi = χ0 ◦Adwi, i = 0, 1, . . . , l.(Note that all wi ∈WΘ fix the subgroups AΘ and χc.) In sum, we have thefollowing description of the A∅-spectrum on (τ, V ):232Lemma 9.2. The restriction of τ to the A∅ is a direct sum of unitarycharacters (with the same multiplicities). LetSpec(A∅, τ) = {χ0, χ1, . . . , χl}be the set of unitary characters of A∅ occurring on (τ, V ), i.e.τ |A∅ =⊕χ∈Spec(A∅,τ)dχ =l⊕i=0dχi,where d is the dimension of the irreducible ◦MΘ0-representation ρ0. Thenthe χi are conjugate to each other by elements in WΘ. More precisely, thereis a element wi ∈WΘ (which may not be unique), such that χi = χ0 ◦Adwi(and one can choose w0 to be identity).Remark 9.3. The explicit description of the wi depends on the concreterepresentation τ , and there is no uniform way to describe them.9.1.2 Formulating an Analogue of Bruhat’s Theorem 7.4We formulate the Theorem 7.4 in Bruhat’s [15]. We keep the settingas in the last subsection, and still assume (τ, V ) to be finite dimensional,irreducible and unitary. Let Spec(A∅, τ) be the finite set of A∅-charactersoccurring in τ as in Lemma 9.2.Definition 9.4. A character χ ∈ Spec(A∅, τ) is called regular ifχ 6= χ ◦Adwfor all w ∈ W −WΘ, i.e. the w-conjugation of χ is not identically equal toχ for all w /∈WΘ.If one (hence all) A∅-characters in Spec(A∅, τ) is regular, we say therepresentation τ is regular.Remark 9.5. If one character in Spec(A∅, τ) is regular in the above sense,then every character in Spec(A∅, τ) are regular. Actually, let χ ∈ Spec(A∅, τ)be a regular character, then the other characters in Spec(A∅, τ) are of theform χ◦Ads−1 for some s ∈WΘ. If the χ◦Ads−1 is not regular, then there isa w ∈W−WΘ such that χ◦Ads−1◦Adw = χ◦Ads−1. Hence χ◦Ad(s−1ws) =χ. But s−1ws is not in WΘ (otherwise w ∈ WΘ a contradiction), thiscontradicts the assumption that χ is regular.233Now we state the theorem in Bruhat’s thesis, which gives a sufficientcondition of irreducibility of unitary parabolic induction of finite dimensionalrepresentations:Theorem 9.6 (Bruhat [15] Theorem 7.4). Let (τ, V ) be a irreducible fi-nite dimensional unitary representation of PΘ (therefore of MΘ), and letSpec(A∅, τ) be the finite set of A∅-character occurring on τ . If one (henceall) character in Spec(A∅, τ) is regular in the sense of Definition 9.4, thenthe normalized parabolic induction IndGP τ is irreducible.Outline of the Proof of Theorem 9.6By the Remark 5.22, we just need to show the Schwartz induction(smooth induction) I satisfies HomG(I, I) = C. By the Remark 5.26, wewill show the irreducibility in by the following two steps:1. show the HomP (V′, (I/I>e)′) = C.2. find a subset Ω ⊂ Θ such that for all w ∈ [WΘ\W/WΩ], w 6= e, theHomPΩ(V′, (IΩ≥w/IΩ>w)′) = {0}.As in Remark 5.26, these two steps correspond to the two cases thatsuppD ⊂ P or suppD * P . The first step is done in the next subsection9.1.3. For the second step, we will choose Ω = ∅ and show the Hom spacesare zero in section 9.2.9.1.3 The Space HomP (V′, (I/I>e)′)In this subsection, we only assume (τ, V ) to be irreducible unitary, with-out assuming it to be finite dimensional.By Lemma 5.24, the space of intertwining distributions with supportscontained in P is linearly isomorphic to the following spaceHomP (V′, (I/I>e)′).By the Theorem 6.1 in Chapter 6, we have the following isomorphism(I/I>e)′ ' I ′e ⊗ U(n−e )(Note that for w = e, the transverse subalgebra n−w is exactly nP .) Also byLemma 4.93, we see the Ie = SIndPPσ is exactly the P -representation (σ,V). Therefore, we have(I/I>e)′ ' V ′ ⊗ U(nP ) (9.2)234(The (I/I>e)′ is isomorphic to a generalized Verma module.)Therefore we have the following inclusionHomP (V′, (I/I>e)′) ' HomP (V ′, V ′ ⊗ U(nP ))⊂ HomM (V ′, V ′ ⊗ U(nP ))where the V ′⊗U(nP ) has the tensor productM -action by the same argumentas in Lemma 8.1. More precisely, we haveHomP (V′, (I/I>e)′) ⊂ HomM (σ̂, σ̂ ⊗ U(nP ))= HomM (τ̂ ⊗ δ−1/2P , τ̂ ⊗ δ−1/2P ⊗ U(nP ))(Here the “hat” means dual representation.)Note that a M = MΘ-equivariant map from τ̂⊗δ−1/2P to τ̂⊗δ−1/2P ⊗U(nP )has to beAΘ-equivariant. The τ restricted toAΘ is a single unitary characterχc of AΘ (the restricted character discussion in the last subsection), whilethe other two components δ−1/2P and U(nP ) are real rational representations.Hence the image of a AΘ-equivariant map from τ̂⊗δ−1/2P to τ̂⊗δ−1/2P ⊗U(nP )must send the τ̂ to τ̂ . Therefore we haveHomM (τ̂ ⊗ δ−1/2P , τ̂ ⊗ δ−1/2P ⊗ U(nP ))⊂ HomM (τ̂ , τ̂)= CRemark 9.7. In the above argument, we don’t need the σ = τ ⊗ δ1/2P to befinite dimensional. We only need the τ to be irreducible unitary, since allsuch representations have restricted AΘ-characters.To summarize the above discussion, we see the HomP (V′, (I/I>e)′) = Cwhen V is an irreducible unitary representation. We have the followingtheorem:Theorem 9.8. Let (τ, V ) be an irreducible unitary representation of M(and P by trivial extension), then the intertwining distributions with supportcontained in P are exactly the scalar intertwining distributions.Remark 9.9. This theorem tells us: even when the parabolic induction isreducible, its non-scalar intertwining distributions are not supported in P .This theorem has the following easy corollary235Corollary 9.10. Let (τ, V ) be an irreducible unitary representation of M(and P by trivial extension), then the normalized Schwartz induction I =SIndGP (τ ⊗ δ1/2P ) is irreducible, if and only if all intertwining distributionshave their supports contained in P . Equivalently, the I is irreducible if andonly if there is no intertwining distribution with its support containing adouble coset other than P .9.2 The Space HomP∅(V′, (I≥w/I>w)′)In this section, we complete the proof of the Theorem 9.6. We stillassume (τ, V ) to be a finite dimensional and irreducible unitary representa-tion.We choose the Ω = ∅, and show the following Hom spaces are zero:HomP∅(V′, (I≥w/I>w)′) = {0}, for all w 6= e, w ∈ [WΘ\W ].In a word, we prove these spaces are zero, by comparing the A∅-Spectrumon V ′ and [(I≥w/I>w)′][n∅•].9.2.1 The Reason to Study n∅-Torsion SubspacesThe condition “(τ, V ) is finite dimensional” is a very strong conditionand it largely simplify the proof of irreducibility of IndGP τ . More precisely,the smooth representation V is nP -trivial, and is mP ∩ n∅-torsion since it isfinite dimensional. Therefore the V is torsion as a n∅-module:V [n∅•] = V,and so is its dual V ′: (V ′)[n∅•] = V ′.For an arbitrary Φ ∈ HomP∅(V ′, (I≥w/I>w)′), since it is P∅-equivariant,it is also n∅-equivariant. In particular, it maps the V ′ = (V ′)[n∅•] to then∅-torsion subspace of (I≥w/I>w)′:Φ ∈ HomC(V ′, [(I≥w/I>w)′][n∅•]).Moreover, since the n∅-torsion subspace is M∅-stable, and the Φ is a M∅-equivariant map from V ′ to [(I≥w/I>w)′][n∅•]. In sum, we have the followingLemmaLemma 9.11. If (τ, V ) is a finite dimensional representation, thenHomP∅(V′, (I≥w/I>w)′) = HomM∅(V′, [(I≥w/I>w)′][n∅•]). (9.3)236By this lemma, we only need to show the right-hand-side is zero, by show-ing the A∅-spectrum on [(I≥w/I>w)′][n∅•] is disjoint from the A∅-spectrumon (τ̂ ⊗ δ−1/2P , V ′). The Chapter 6, 7 and 8 are devoted to the study ofA∅-spectrum on [(I≥w/I>w)′][n∅•]:• In Chapter 6, we have shown(I≥w/I>w)′ ' I ′w ⊗ U(n−w), as M∅-spaces.• In Chapter 7, we have shown (under the above isomorphism)[I ′w ⊗ U(n−w)][n∅•] = [I ′w][n∅•] ⊗ U(n−w).• In Chapter 8, we use Shapiro’s Lemma to compute the M∅-structure(hence also the A∅-spectrum) on [I ′w][n∅•].9.2.2 Comparison of the A∅-SpectrumsWe compare the A∅-Spectrums on V ′ and [(I≥w/I>w)′][n∅•]. Given anarbitrary Φ ∈ HomM∅(V ′, [(I≥w/I>w)′][n∅•]), by the Lemma 9.11 and themain theorems in Chapter 6, 7, the Φ is reduced to a M∅-equivariant mapΦ : V ′ → (I ′w)[n∅•] ⊗ U(n−w).Since the V ′ is finite dimensional, the image of Φ is thus contained in(I ′w)[n∅k] ⊗ Un(n−w)for some k > 0 and n ≥ 0 (large enough). To show theHomM∅(V′, [(I≥w/I>w)′][n∅•])is {0} or equivalently the above arbitrary Φ is zero, we just need to showHomM∅(V′, (I ′w)[n∅k] ⊗ Un(n−w)) = {0}for all k > 0, n ≥ 0. This requires us to compare the A∅-spectrum on V ′and (I ′w)[n∅k] ⊗ Un(n−w).The A∅-spectrum on V ′ is the following finite set of A∅-characters:Spec(A∅,̂τ ⊗ δ1/2P ) = {χ−1 ⊗ δ−1/2P : χ ∈ Spec(A∅, τ)} (9.4)We are left to study the A∅-spectrum on (I ′w)[n∅k] ⊗ Un(n−w).237The A∅-Spectrum on (I ′w)[n∅k] ⊗ Un(n−w)By Lemma 8.1, the tensor product (I ′w)[n∅k]⊗Un(n−w) is actually a tensorproduct of M∅-representations hence also A∅-representations. We just needto find the A∅-spectrum on the annihilator (I ′w)[n∅k].The M∅-action on (I ′w)[n∅k] is studied in Chapter 8 and summarized inLemma 8.27, i.e. the M∅ acts on (I ′w)[n∅k] by the representation[(τ ⊗ δ1/2P )w ⊗ ηk]∧ ⊗ γ = τ̂w ⊗̂(δ1/2P )w ⊗ η̂k ⊗ γ.(Note that the σ = τ ⊗ δ1/2P in the concrete case.) In particular, by re-stricting the M∅-action to the A∅, we have the following description of theA∅-spectrum:Lemma 9.12. The A∅-spectrum on (I ′w)[n∅k] consists of A∅-characters ofthe following form:(χw)−1 ⊗ (δ−1/2P )w ⊗ δw ⊗ µwhere χ ∈ Spec(A∅, τ), δw is the A∅-character in (8.23), and µ is a A∅-character on the η̂k, of the following form∏α∈Σ+α−kα (9.5)where kα are non-negative integers such that∑kα = j for some j ≤ k − 1(The µ comes from the A∅-spectrum on ηk).The A∅-spectrum on the transverse derivative space Un(n−w) consists ofA∅-characters of the form ∏α∈Σ−α∈w−1(Σ−−Σ−Θ)αlα (9.6)where lα are non-negative integers such that∑lα ≤ n.In sum, we have:Lemma 9.13. The A∅-spectrum on the (I ′w)[n∅k] ⊗ Un(n−w) consists of A∅-characters as follows:(χw)−1 ⊗ (δ−1/2P )w ⊗ δw ⊗ µ⊗ ν (9.7)where χ ∈ Spec(A∅, τ), δw is the A∅-character in (8.23), µ is the A∅-character as in (9.5) and ν is the A∅-character as in (9.6).238The Spectrum Spec(A∅, τw) of Regular τRecall that in Definition 9.4, the τ is called regular, if any A∅-characteroccurring in τ is regular. Given the (τ, V ) and an element w ∈W , we denotebyτw = τ ◦Adwthe twisted representation of w−1MPw on V . And similarly, for a characterχ of A∅, we denote by χw the A∅-character χ ◦Adw.For a regular τ , we have the following lemma about the A∅-spectrum ofτw for w /∈WΘ:Lemma 9.14. Suppose the finite dimensional irreducible unitary represen-tation τ is regular, and w ∈ W −WΘ. Then the finite set Spec(A∅, τw) isdisjoint from Spec(A∅, τ).Proof. Obviously, one hasSpec(A∅, τw) = {χw : χ ∈ Spec(A∅, τ)}.Suppose a χw ∈ Spec(A∅, τw) is also in Spec(A∅, τ), then there is a s ∈WΘsuch that χw = χs. Then one has χws−1= χ. Since τ is regular, so is χ, thisimplies ws−1 ∈WΘ, which further implies w ∈WΘ, a contradiction.Comparison of The A∅-Spectrum and Proof of Bruhat’s Theorem7.4SinceHomM∅(V′, (I ′w)[n∅k] ⊗ Un(n−w)) ⊂ HomA∅(V ′, (I ′w)[n∅k] ⊗ Un(n−w)),to show the first M∅-Hom space is zero, we just need to show the second A∅-Hom space is zero. In (9.4) and (9.7), we have written down the A∅-spectrumon V ′ and (I ′w)[n∅k] ⊗ Un(n−w). We just need to show every A∅-characteroccurring on V ′ cannot occur on (I ′w)[n∅k]⊗Un(n−w). For simplicity, we writethe characters additively.We just need to show the A∅-character of the form −χ − 12δP and theA∅-character of the form −w−1χ′ − 12w−1δP + δw + µ + ν are never equal.(Here χ, χ′ are A∅-characters in Spec(A∅, τ) which may not be equal.)Assume these two A∅-characters are equal:−χ− 12δP = −w−1χ′ − 12w−1δP + δw + µ+ ν.239Then we must haveχ = w−1χ′ (9.8)−12δP = −12w−1δP + δw + µ+ νThis is because the χ and w−1χ′ are unitary characters with values on theunit circle, and all the other characters are real characters taking values inR>0. Therefore for the two A∅-characters to be equal, their pure imaginaryand real parts have to be equal respectively.Now since the w is a non-identity element in the representative set[WΘ\W ], it is not in the subgroup WΘ. If τ is regular as in Bruhat’s theorem,by Lemma 9.14, we see the χ and w−1χ′ cannot be equal. Therefore everyA∅-character occurring on (̂τ ⊗ δ1/2P , V ′) cannot occur in (I ′w)[n∅k]⊗Un(n−w),and the proof is completed.9.3 Minimal Principal SeriesIn this section, we prove our analogous result (Theorem 9.15) of Bruhat’sTheorem 7.2a in [15]. In this section, we letP = P∅be the minimal parabolic subgroup (i.e. Θ = ∅, then the only choice of Ωis the empty set). The representative set [WΘ\W ] is exactly W , which is inone-to-one correspondence with the (P∅, P∅)-double cosets.• In 9.3.1, we formulate and prove our analogue (Theorem 9.15) of theTheorem 7.2a in [15].• In 9.3.2, we study the real parts of the A∅-spectrum on [I ′w][n∅•] ⊗U(n−w). We will show the real parts ofA∅-spectrum on V ′ and [I ′w][n∅k]⊗Un(n−w) agree only when k = 1, n = 0. Then for certain special cases,(e.g. split groups), we can actually write down the local intertwiningdistributions as integrations by Shapiro’s Lemma.9.3.1 An Analogue of Bruhat’s Theorem 7.2aAs above, let P = P∅ be the minimal parabolic subgroup. In the Lang-lands decomposition ◦M∅A∅N∅, the component ◦M∅ is a compact group,240hence the irreducible unitary representation (τ, V ) of M∅ = ◦M∅ × A∅ hasto be finite dimensional.We now formulate the analogue of Theorem 7.2a in [15]:Theorem 9.15. Let P = P∅ be the minimal parabolic as above. If therepresentation τw = τ ◦ Adw is not equivalent to τ for all w ∈ W,w 6= e,then the induced representation IndGP∅τ is irreducible.Proof. Similar to the proof of Theorem 9.6, we just need to showHomP∅(V′, (I≥w/I>w)′) = {0}for all w 6= e.Since V ′ is finite dimensional, we haveHomP∅(V′, (I≥w/I>w)′) = HomM∅(V′, [(I≥w/I>w)′][n∅•])i.e. the image of each Φ ∈ HomP∅(V ′, (I≥w/I>w)′) has its image in thetorsion subspace. Combining the results in Chapter 6 and 7, we haveHomM∅(V′, [(I≥w/I>w)′][n∅•]) = HomM∅(V′, [I ′w][n∅•] ⊗ U(n−w)).Again, since V ′ is finite dimensional, there exists k > 0, n ≥ 0 (large enough),such thatHomM∅(V′, [I ′w][n∅•] ⊗ U(n−w)) = HomM∅(V ′, [I ′w][n∅k] ⊗ Un(n−w)).To summarize the above steps, we haveHomP∅(V′, (I≥w/I>w)′) = HomM∅(V′, [(I≥w/I>w)′][n∅•])= HomM∅(V′, [I ′w][n∅•] ⊗ U(n−w))= HomM∅(V′, [I ′w][n∅k] ⊗ Un(n−w))We just need to show HomM∅(V′, [I ′w][n∅k] ⊗ Un(n−w)) = {0} by comparingthe M∅-actions on V ′ and [I ′w][n∅k] ⊗ Un(n−w).On the V ′, the M∅ acts by τ̂ ⊗ δ−1/2P , while on the [I ′w][n∅k]⊗Un(n−w), theM∅ acts by τ̂w⊗w−1δ−1/2P ⊗ η̂k⊗γw⊗U(n−w). Let Φ ∈ HomM∅(V ′, [I ′w][n∅k]⊗Un(n−w)) be an arbitrary element. Since the τ̂ is unitary, the image of Φhas to lay inside the τ̂w. This is because Φ is also A∅-equivariant, the onlyunitary A∅-eigensubspace in [I ′w][n∅k] ⊗ Un(n−w) is τ̂w. Therefore, the Φ isinside HomM∅(τ̂ , τ̂w). By Schur’s Lemma, this space is zero, if τ satisfiesthe condition τw 6= τ for all w 6= e. Therefore the Φ has to be zero, andHomM∅(V′, [I ′w][n∅k] ⊗ Un(n−w)) = {0}.2419.3.2 Real Parts of the A∅-SpectrumIn the proof of Bruhat’s theorem 7.4, we show the A∅-characters thatoccurring on V ′ cannot occur on (I ′w)[n∅k] ⊗ Un(n−w), by showing their pureimaginary parts are never equal on all the other double cosets other than P .Actually, the real parts of the A∅-characters occurring on (I ′w)[n∅k]⊗Un(n−w)also give us interesting results. We will study the real parts of the A∅-spectrum on (I ′w)[n∅k]⊗Un(n−w), for P = P∅ the minimal parabolic subgroup.The real parts of the A∅-characters occurring on (I ′w)[n∅k] ⊗ Un(n−w) areof the form:−12w−1δP + δw + µ+ ν,where δw, µ, ν are A∅-character as follows:δw = −∑α∈Σ+α∈w−1(Σ−−Σ−Θ)mααµ = −∑α∈Σ+kααν =∑α∈Σ−α∈w−1(Σ−−Σ−Θ)lααHere all characters are written additively, mα are the multiplicity of the(restricted) root α, kα are non-negative integers such that∑kα ≤ k− 1, lαare non-negative integers such that∑lα ≤ n. One can see the δw, µ, ν areall integral combinations of negative roots.The real parts of A∅-spectrum on V ′ are all equal to −12δP , we may askwhen does this equal to the real parts −12w−1δP + δw + µ+ ν.Lemma 9.16. For all w ∈W , one has− 12δP = −12w−1δP + δw. (9.9)Proof. This is elementary. Note that we have assumed P = P∅, thereforeδP =∑α∈Σ+mααδw = −∑α∈Σ+α∈w−1Σ−mαα242(Note that the δw depends on w.)First we decomposeΣ+ = {α ∈ Σ+ : w−1α ∈ Σ+}∐{α ∈ Σ+ : w−1α ∈ Σ−}.Then we have12w−1δP =12w−1∑α∈Σ+w−1α∈Σ+mαα+12w−1∑α∈Σ+w−1α∈Σ−mαα=12∑α∈Σ+wα∈Σ+mαα+12∑α∈Σ−wα∈Σ+mααThen12w−1δP − 12δP =12∑α∈Σ+wα∈Σ+mαα+12∑α∈Σ−wα∈Σ+mαα− 12∑α∈Σ+mαα=12∑α∈Σ−wα∈Σ+mαα− 12∑α∈Σ+wα∈Σ−mαα=12∑α∈Σ−α∈w−1Σ+mαα− 12∑α∈Σ+α∈w−1Σ−mαα= −12∑α∈Σ+α∈w−1Σ−mαα− 12∑α∈Σ+α∈w−1Σ−mαα= −∑α∈Σ+α∈w−1Σ−mαα= δwBy this Lemma, we see if the real parts of A∅-characters occurring in V ′and (I ′w)[n∅k] ⊗ Un(n−w) are equal, i.e −12δP = −12w−1δP + δw + µ + ν, thenwe must haveµ = 0, ν = 0.Equivalently, this could only happen when k = 1, n = 0. Recall that the µcomes from the η̂k = F∗k which is only non-trivial on higher annihilators, and243the ν comes from the transverse derivatives in U(n−w). Then µ = 0, ν = 0implies the distribution has neither higher annihilators nor transverse deriva-tives. This means the image of Φ ∈ HomM∅(V ′, (I ′w)[n∅k] ⊗Un(n−w)) is inside(I ′w)[n∅1].Therefore, we haveHomP∅(V′, (I≥w/I>w)′) = HomM∅(V′, (I ′w)[n∅1]) (9.10)= HomM∅(V′, H0(n∅, I ′w))= HomM∅(τ̂ ⊗ δ−1/2P , τ̂w ⊗ w−1δ−1/2P ⊗ γw)(The last step is by Shapiro’s Lemma.)Although we don’t have any immediate consequence, at least we knowthe local intertwining distributions do not have any transverse derivativesin their expression, and these local intertwining distributions are given byintegrations on the quotient spaces N∅ ∩ w−1P∅w\N∅.244Chapter 10General Results and FutureWorkIn this chapter, we sketch the proof of the general results (for arbitraryΩ ⊂ Θ), and discuss the topics we will study in the future.10.1 Generalization of Chapter 6We keep the setting as in the introductory subsection 1.1.3, e.g. thenotations G,P = PΘ, G, P = PΘ, τ, σ = τ⊗δ1/2P , I = SIndGPσ have the samemeaning as there. For each subset Ω of Θ, and w in the set [WΘ\W/WΩ] ofminimal representatives, we letGΩw = PwPΩGΩ≥w =∐PwPΩ⊂PxPΩPxPΩGΩ>w = GΩ≥w −GΩwand let IΩw , IΩ≥w, IΩ>w be their corresponding local Schwartz inductions.10.1.1 Formulating the TheoremAs in Chapter 6, the dual quotient (IΩ≥w/IΩ>w)′ is exactly the kernel ofthe following restriction mapResΩw : (IΩ≥w)′ → (IΩ>w)′.LettΩw := nΩ ∩ w−1nPw = nΩ ∩ w−1nΘwbe the transverse subalgebra.As in the 6.1.1, since GΩ≥w is open in G and GΩw is closed in GΩ≥w, theU(g)-derivatives of distributions on GΩw are supported in GΩw (vanishing onGΩ>w). As a generalization of Theorem 6.1, we have the following theorem:245Theorem 10.1. We have the following linear maps as in (6.1) and (6.2):S(GΩw, V )′ ⊗ U(tΩw)→ Ker{S(GΩ≥w, V )′ → S(GΩ>w, V )′}(IΩw )′ ⊗ U(tΩw)→ Ker{(IΩ≥w)′ → (IΩ>w)′}i.e. they are given by U(tΩw)-derivatives of distributions on GΩw. The abovetwo maps are isomorphisms.10.1.2 Sketch of the ProofThe key points of the proof are• Change of neighoubrhood: Lemma 4.38 and Lemma 4.80.• The Zw is a tubular neighbourhood of GΩw:Zw ' GΩw × (NΩ ∩ w−1NPw).• The tensor product property, i.e. (E-6) in Proposition 4.30.• The distribution with point support, i.e. the kernel of the restrictionmapS(NΩ ∩ w−1NPw, V )′ → S(NΩ ∩ w−1NPw − {e}, V )′is exactly the enveloping algebra U(nΩ ∩ w−1nPw) = U(tΩw).The GΩw and the Tubular Neighbourhood ZwAs in 6.1.2, we letZw = PNPw = PΘNΘwfor all w ∈W . This is an Zariski open subset of G, and the Lemma 6.3 stillholds.For each Ω ⊂ Θ and w ∈ [WΘ\W/WΩ], the GΩw = PwPΩ is isomorphicto P × {w} × PΩPΩ∩w−1Pw . The canonical map(MΩ ∩ w−1NPw) · (NΩ ∩ w−1NPw)→ PΩ ∩ w−1Pw\PΩ (10.1)is an isomorphism of manifolds and real algebraic varieties. Similar to theLemma 8.8, we have the following Lemma on the structure of GΩw:246Lemma 10.2. The mapP × (MΩ ∩ w−1NPw) · (NΩ ∩ w−1NPw)→ GΩw = PwPΩ (10.2)(p,mn) 7→ pwmnis an isomorphism of manifolds and real algebraic varieties. The GΩw is anonsingular closed subvariety of Zw and a closed regular submanifold of Zw.The Zw is a tubular neighbourhood of GΩw:Zw ' P × w−1NPw ' GΩw × (NΩ ∩ w−1NPw). (10.3)As in 6.1.3, by the Lemma 4.38 and 4.80, we have the following isomor-phisms between kernels of restriction maps of distributions:Ker{S(GΩ≥w, V )′ → S(GΩ>w, V )′} ' Ker{S(Zw, V )′ → S(Zw −GΩw, V )′}Ker{(IΩ≥w)′ → (IΩ>w)′} ' Ker{(SIndZwP σ)′ → (SIndZw−GΩwP σ)′}Distributions on Schwartz Function SpacesWe first show the following isomorphism on distributions on Schwartzfunctions:S(GΩw, V )′ ⊗ U(tΩw) ∼−→ Ker{S(Zw, V )′ → S(Zw −GΩw, V )′}.First by the above Lemma 10.2 and the (E-6) of Proposition 4.30, wehaveS(Zw, V ) ' S(GΩw, V ) ⊗̂ S(NΩ ∩ w−1NPw,C)S(Zw −GΩw, V ) ' S(GΩw, V ) ⊗̂ S(NΩ ∩ w−1NPw − {e},C)Since all spaces are nuclear, we take the strong dual of the above isomor-phisms, and we see the Ker{S(Zw, V )′ → S(Zw − GΩw, V )′} is isomorphictoS(GΩw, V )′ ⊗̂Ker{S(NΩ ∩ w−1NPw,C)′ → S(NΩ ∩ w−1NPw − {e},C)}.By the Lemma 6.7, we see the above space is exactlyS(GΩw, V )′ ⊗ U(nΩ ∩ w−1nPw) = S(GΩw, V )′ ⊗ U(tΩw).247Distributions on Schwartz Induction SpacesWe show the isomorphism(IΩw )′ ⊗ U(tΩw) ∼−→ Ker{(SIndZwP σ)′ → (SIndZw−GΩwP σ)′}.By the Lemma 4.91, we haveSIndZwP σ' SIndPPσ ⊗̂ S((MΩ ∩ w−1NPw) · (NΩ ∩ w−1NPw),C)⊗̂ S(NΩ ∩ w−1NPw,C)' SIndGΩwP σ ⊗̂ S(NΩ ∩ w−1NPw,C)SIndZw−GΩwP σ' SIndPPσ ⊗̂ S((MΩ ∩ w−1NPw) · (NΩ ∩ w−1NPw),C)⊗̂ S(NΩ ∩ w−1NPw − {e},C)' SIndGΩwP σ ⊗̂ S(NΩ ∩ w−1NPw − {e},C)Since all spaces are nuclear, by taking their dual, we see the kernelKer{(SIndZwP σ)′ → (SIndZw−GΩwP σ)′} is isomorphic to(SIndGΩwP σ)′ ⊗̂Ker{S(NΩ ∩ w−1NPw,C)′ → S(NΩ ∩ w−1NPw − {e},C)},which is exactly (IΩw )′ ⊗ U(tΩw).10.2 Generalization of Chapter 710.2.1 Formulating the TheoremThe following theorem is a generalization of the main result (Theorem7.1) in Chapter 7:Theorem 10.3. The nΩ-torsion subspace on the kernels of the restrictionmap S(GΩ≥w, V )′ → S(GΩ>w, V )′ and (IΩ≥w)′ → (IΩ>w)′ are given by[S(GΩw, V )′ ⊗ U(tΩw)][n•Ω] = [S(GΩw, V )′][n•Ω] ⊗ U(tΩw)[(IΩw )′ ⊗ U(tΩw)][n•Ω] = [(IΩw )′][n•Ω] ⊗ U(tΩw)248By the same argument as in 7.1.1, we just need to show the first equality.by the same argument as in 7.1.2, we regard all distribution spaces as rightmodules over enveloping algebras.We use the same notation as in Chapter 7, letU = S(GΩw, V )′K = S(GΩw, V )′ ⊗ U(tΩw)M = [S(GΩw, V )′][n•Ω] ⊗ U(tΩw)For each n ≥ 0, we letKn = S(GΩw, V )′ ⊗ Un(tΩw)Mn = [S(GΩw, V )′][n•Ω] ⊗ Un(tΩw)We have K0 = U ,M0 = U[n•Ω]. We need to showM = K[n•Ω].10.2.2 Sketch of the ProofThe Easy Part M⊂ K[n•Ω]We proveMk ⊂ K[n•Ω]by induction on k. The case k = 0 is obvious. Assuming Mk−1 ⊂ K[n•Ω], weshow Mk ⊂ K[n•Ω]. By the same argument as in 7.2.2, we just need to showthe elements of the following form are in the torsion subspace K[n•Ω]:Φ · Y1 · · ·Ykwhere Φ ∈ U[n•Ω], Y1, . . . , Yk ∈ tΩw.By Lemma 7.20, we just need to show there exists a large n such thatΦ · Y1 · · ·Yk · (nnΩ) ⊂ K[n•Ω].Actually, by writingΦ · Y1 · · ·Yk = (Φ · Y1 · · ·Yk−1) · Yk,we see there exists a n1 ≥ 0, such that (Φ · Y1 · · ·Yk−1) · (nn1Ω ) = {0} sincethe Φ ·Y1 · · ·Yk−1 is in the nΩ-torsion subspace by the induction hypothesis.Also there exists a n2 ≥ 0 large enough such that[. . . [[Yk, X1], X2] . . . Xn2 ] ∈ nΩ,249for all Xi ∈ nΩ.Therefore let n = n1 +n2 + 1, and use the formula in Lemma 7.8, we seeΦ · Y1 · · ·Yk · (nnΩ) ⊂ K[n•Ω].Linear Order on PBW-Basis and the Linear Filtration of KAs in 7.3, we fix a basis {Y1, . . . , Yd} of tΩw consisting of root vectors andsatisfyingHt(Y1) ≤ Ht(Y2) ≤ . . . ≤ Ht(Yd).Recall that if Yi is a root vector of the restricted root α, and α is uniquelywritten as non-positive integral combinations of simple roots α1, . . . , αr:α = −r∑i=1niαi, ni ≥ 0,then the height of Yi is defined as Ht(Yi) =∑ri=1 ni.With the above labeling on the basis of tΩw, we thus have a PBW-basisof the enveloping algebra{Y I : I ∈ Ld}where Ld = Zd≥0 is the set of multi-index, and for a I = (i1, i2, . . . , id), the Y Imeans the product Y i11 Yi22 · · ·Y idd in U(tΩw) and the Y I form a PBW-basis.As in 7.3.1, we choose the linear order on the index set Ld thus the abovelabeling of Y1, . . . , Yd gives a linear order on the PBW-basis YI of U(tΩw).As in 7.3.6, for each multi-index I ∈ Ld, let I− be its lower adjacent (seeDefinition 7.43), and letUI(tΩw) = the subspace of U(tΩw) spanned by YJ , J ≤ I,and letKI = U ⊗ UI(tΩw).Then the {KI : I ∈ Ld} form an exhaustive filtration of K.The Decompositions of Vector FieldsAs in 7.1.5, given an element X ∈ g (complexified Lie algebra), itscorresponding left invariant vector field is denoted byXL, and right invariantvector field is denoted by XR, and for a point x ∈ G, the tangent vector ofthe vector field XL (resp. XR) at x is denoted by XLx (resp. XRx ).250LetLtΩwx := spanR{XLx : X ∈ tΩw}.This is a subspace of the tangent space TxG = TxZw, and one hasTxG = TxZw = TxGΩw ⊕ LtΩwxfor all x ∈ GΩw, i.e. the subalgebra tΩw is transverse to the submanifold GΩwat every point on it.Given an element H ∈ g, let HL be the corresponding left invariantvector field on G, let HL|GΩw be the restriction of the vector field to thesubmanifold GΩw, then this restriction is uniquely written as(HL|GΩw)x =k∑i=1Ai(x)XRi,x +l∑i=1Bi(x)ZLi,x +d∑i=1Ci(x)Y Li,xwhere {X1, . . . , Xk} is an arbitrary basis of p, {Z1, . . . , Zl} is an arbitrarybasis of mΩ ∩ w−1pw + nΩ ∩ w−1pw, and {Y1, . . . , Yd} is an arbitrary basisof tΩw. The Ai, Bi, Ci are algebraic functions on the variety GΩw.By the same argument as in Lemma 7.40, we can showLemma 10.4. Let Y1, . . . , Yd be a basis of tΩw with non-decreasing heights.Let X ∈ nΩ be an arbitrary element, and let [Yj , X] be the Lie algebra bracket.Then in the above decomposition of [Yj , X]L|GΩw , we have all Ck ≡ 0 on GΩw,for all k ≥ j.The KI are nΩ-SubmodulesThe most crucial results to prove the inclusionM⊃ K[n•Ω] is: each KI isnΩ-stable, hence is a nΩ-submodule of K. Namely we haveLemma 10.5. For each multi-index I ∈ Ld, the subspace KI of the rightnΩ-module K is stable under the right multiplication of nΩ.The quotient space KI/KI− is isomorphic to U as nΩ-modules.As in the Remark 7.45, we just need to show the following element is inKI− :Φ · [Yj , X] · Y I−Jwhere Φ ∈ U , X ∈ nΩ and the multi-index J = (0, . . . , 0, 1, 0, . . . , 0) (withjth entry equal to 1 and all other entries zero). This is easy to see by theabove Lemma on the coefficients Ci and the Lemma 7.41.251The Hard Part M⊃ K[n•Ω]To show the inclusion K[n•Ω] ⊂M, we just need to showKI[n•Ω] ⊂Mby induction on the linear order of I. The case I = (0, . . . , 0) is clear. Foran arbitrary I, let I− be its lower adjacent (Definition 7.43), and we assumeKI−[n•Ω] ⊂M.For an arbitrary element∑J≤I ΦJ · Y J of KI[n•Ω], first its image in thequotient KI/KI− is exactly ΦI · Y ImodKI− . Since the original element isnΩ-torsion, we see the image ΦI ·Y ImodKI− is also nΩ-torsion. The quotientKI/KI− is isomorphic to U as nΩ-modules, therefore the ΦI ∈ U has to benΩ-torsion.By the (easy part) inclusionM⊂ K[n•Ω], we see the leading term ΦI · Y Iis in K[n•Ω]. Therefore the tail sum is also torsion:∑J<IΦJ · Y J ∈ K[n•Ω].By the induction hypothesis the tail term∑J<I ΦJ · Y J is in M, and theleading term ΦI · Y I is in M (since ΦI is torsion), therefore the entire sumis in M: ∑J≤IΦJ · Y J ∈M.10.3 Generalization of Chapter 8We keep the notations and setting as the last two sections. We cangeneralize some of the results in Chapter 8 to arbitrary subset Ω ⊂ Θ. Buta crucial problem is: the generalization of Lemma 8.1 is not true.The Lemma 8.1 says, the isomorphism (I≥w/I>w)′ ' I ′w ⊗U(n−w) is M∅-equivariant when the right-hand-side is endowed with the tensor productM∅-action. However for the general Ω, the tensor product (IΩw )′ ⊗ U(tΩw) isnot a tensor product of MΩ-representations, as one can see the subalgebratΩw = nΩ ∩ w−1nΘw is not stable under MΩ-conjugation.However, to make the results as general as possible, we state the gen-eralizations some results in Chapter 8. We omit the proof since they areproved by exactly the same way as the corresponding Lemma in Chapter 8252Lemma 10.6 (Annihilator-Invariant Trick). The kth annihilator space on(IΩw )′ is isomorphic to[(IΩw )′][nkΩ] ' H0(nΩ, (IΩw ⊗ FΩk )′),where FΩk = U(nΩ)/(nkΩ) is a finite dimensional representation of NΩ (orPΩ).This is prove exactly the same way as in 8.1.2, by algebraic tricks.Lemma 10.7. The local Schwartz induction IΩw = SIndGΩwP σ is isomorphicto the Schwartz inductionSIndPΩPΩ∩w−1Pwσwwhere σw = σ ◦ Adw is the representation of w−1Pw and regarded as arepresentation of PΩ ∩ w−1Pw by restriction.This is proved by the same way as Lemma 8.9 since both spaces are(linearly) isomorphic to the Schwartz function space S((MΩ ∩ w−1NPw) ·(NΩ ∩ w−1NPw), V ).Lemma 10.8 (Tensor Product Trick). The tensor product IΩw ⊗ FΩk is iso-morphic toSIndPΩPΩ∩w−1Pw(σw ⊗ FΩk ).This is proved by the same way as Lemma 8.20.10.4 Future WorksAs mentioned in the section 1.2, some topic of our future work are:• Irreducibility: Reproduce the result in [40] about complex groups.• Globalization of local intertwining distributions, namely, findthe original D ∈ HomP (I, V ) from its restrictions to open subsets.(e.g. for SL(3,C), reproduce all intertwining operators studied in [37]).However in this final section, we want to show our plan on how to gener-alize the methods to study the parabolic inductions of infinite dimensionalrepresentations.As we have seen in Chapter 9, the condition “V is finite dimensional”largely simplify the proof: the entire V ′ is n∅-torsion and the image of an253arbitrary Φ ∈ HomP∅(V ′, (I≥w/I>w)′) is contained in the n∅-torsion subspaceof (I≥w/I>w)′. Then we haveHomP∅(V′, (I≥w/I>w)′) = HomM∅(V′, [(I≥w/I>w)′][n∅•])and we just need to study the M∅-structure on [(I≥w/I>w)′][n∅•].In general, when the V is infinite dimensional, the V and V ′ are only nP -invariant (or nP -torsion) if we assume (τ, V ) to be irreducible. Therefore,instead of considering n∅-torsion subspaces, we need to study nP -torsionsubspaces. Unlike in Chapter 6, 7 and 8, we will choose the subset Ω = Θ.For simplicity, we still let(τ, V ) = an irreducible unitary representation of P.As shown in Theorem 9.8, the interesting phenomenon only occur on non-identity double cosets. To show the irreducibilities of I = SIndGPσ whereσ = τ ⊗ δ1/2P , we need to showHomP (V′, (IΘ≥w/IΘ>w)′) = {0}.As in the main body of the thesis, we are required to study the quotientdual (IΘ≥w/IΘ>w)′ and its nP = nΘ-torsion subspace.10.4.1 A ConjectureFor each w ∈ [WΘ\W/WΘ], by applying the Theorem 10.1, we have thefollowing isomorphism(IΘ≥w/IΘ>w)′ ' (IΘw )′ ⊗ U(tΘw).By the Theorem 10.3, we can identify its nP = nΘ-torsion subspace:[(IΘw )′ ⊗ U(tΘw)][n•P ] = [(IΘw )′][n•P ] ⊗ U(tΘw).However, as we have seen above or in the introductory subsection 1.1.3,the first obstacle we meet is: the above tensor product is not a tensorproduct of MΘ-representations as the tΘw = nP ∩w−1nPw is not stable underMΘ = MP conjugation.Fortunately, if the w normalizes MP (or equivalently normalizing thesubsystem 〈Θ〉 spanned by Θ), the tΘw is stable under the MP -conjugation,and the above tensor product is indeed a tensor product of representationsof MP .As suggested by all examples we have known, we propose the followingconjecture:254Conjecture 10.9. If w does not normalize the MP = MΘ, then we haveHomMP (V′, (IΘ≥w/IΘ>w)′) = {0}.10.4.2 Another Conjecture of CasselmanAs we have seen in the introductory subsection 1.1.3, the second obstaclewe meet is: we cannot apply Shapiro’s lemma to the Schwartz inductionSIndPP∩w−1Pw(σw ⊗ FΘk ).The main difficulty is: the P and P ∩ w−1Pw are neither unimodularnor cohomologically trivial. More importantly, the quotientP ∩ w−1Pw\Pis not NP -transitive. Therefore we cannot apply Shapiro’s Lemma directly.To overcome this obstacle, Casselman has proposed a conjecture, which willbe formulated below.Let PΘ∩wΘ be the standard real parabolic subgroup corresponding tothe subset Θ ∩ wΘ. It is contained in PΘ, and we denote its image underthe quotient mapPΘ →MΘ ' PΘ/NΘby Qw. This Qw is a parabolic subgroup in MΘ = MP , and let MQwNQwbe its Levi decomposition.For the representation (ξ,H) of MΘ = MP , let (ξ∞, H∞) be its Harish-Chandra module. By a theorem of Hecht-Schmid, the quotient spaceH∞/nQwH∞is a finitely generated Harish-Chandra module of MQw , we denote its Harish-Chandra globalization (in the sense of Casselman’s [17]) by(ρ, U).The canonical map (of Harish-Chandra modules) V∞ → V∞/nQwV∞ ex-tends to a map(ξ,H)→ (ρ, U)and this map is a MQw -map.We can then define the following map by integrationSIndPΘPΘ∩w−1PΘw(ξw)→ SIndMΘw−1MQww(ρw).255This map induces a map[SIndMΘw−1MQww(ρw)]′ → H0(nP , [SIndPΘPΘ∩w−1PΘw(ξw)]′).Casselman proposed the following conjecture:Conjecture 10.10 (Casselman). The above map is an isomorphism.By applying this conjecture to (ξ,H) = (σw ⊗ FΘk , V ⊗ FΘk ), we mightbe able to compute the MP = MΘ-action on the H0(nP , (IΘw ⊗ FΘw )′). Butwe are still working on this conjecture.256Bibliography[1] Avraham Aizenbud and Dmitry Gourevitch. Schwartz functions onNash manifolds. International Mathematics Research Notices, 2008-1:rnm155, January 2008.[2] Avraham Aizenbud and Dmitry Gourevitch. 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Schwartz analysis and intertwining distributions Liu, Xinyu 2017
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Title | Schwartz analysis and intertwining distributions |
Creator |
Liu, Xinyu |
Publisher | University of British Columbia |
Date Issued | 2017 |
Description | In this dissertation, we combine the work of A. Aizenbud and D. Gourevitch on Schwartz functions on Nash manifolds, and the work of F. du Cloux on Schwartz inductions, to develop a toolbox of Schwartz analysis. We then use these tools to study the intertwining operators between parabolic inductions, and study the behavior of intertwining distributions on certain open subsets. Finally we use our results to give new proof of results in Bruhat's thesis on irreducibilities of degenerate principal series and minimal principal series. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2017-12-22 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0362407 |
URI | http://hdl.handle.net/2429/64171 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2018-02 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
Aggregated Source Repository | DSpace |
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