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Arithmetic theory of symmetrizable split maximal Kac-Moody groups Abbaspour Tazehkand, Hesameddin 2012

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Arithmetic Theory of Symmetrizable Split Maximal Kac-Moody Groups by Hesameddin Abbaspour Tazehkand  B.Sc., Sharif University of Technology, 2004 M.Sc., Simon Fraser University, 2006  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Mathematics)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) February 2013 © Hesameddin Abbaspour Tazehkand 2012  Abstract In this thesis we present a reduction theory for the symmetrizable split maximal Kac-Moody groups. However there are many technical difficulties before one can even formulate a reduction theorem. Combining the two main approaches commonly seen in the literature we define a group, first over any field of characteristic zero and then on any commutative ring of characteristic zero. Then we prove a number of structural properties of the group such as representation in the highest weight modules, existence of a Tits system and an Iwasawa decomposition over R and C. Finally we arrive at reduction theory which can only hold for part of the group.  ii  Table of Contents Abstract  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vii  1  2  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  1.1  The Aim  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  1.2  Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . .  1  1.3  Notation  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2  Kac-Moody Algebras: A Primer . . . . . . . . . . . . . . . . . . . .  4  2.1  Definitions  4  2.2  A Classification of GCMs  2.3  Root System  2.4  An Analogue of the Killing Form  . . . . . . . . . . . . . . . . .  9  2.5  Integrable Modules . . . . . . . . . . . . . . . . . . . . . . . . .  10  2.6  Weyl Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  12  2.6.1  Definition  . . . . . . . . . . . . . . . . . . . . . . . . .  12  2.6.2  Action on the Weight Space . . . . . . . . . . . . . . . .  14  2.6.3  W as a Coxeter Group . . . . . . . . . . . . . . . . . . .  15  Geometry of the Weyl Group . . . . . . . . . . . . . . . . . . . .  15  2.7.1  Real and Imaginary Roots . . . . . . . . . . . . . . . . .  15  2.7.2  The Tits Cone  . . . . . . . . . . . . . . . . . . . . . . .  15  Kac-Moody Algebras: Highest Weight Modules . . . . . . . . . . .  18  3.1  18  2.7  3  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  5  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  7  Verma Modules . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  Table of Contents  4  3.2  The Irreducible Quotient . . . . . . . . . . . . . . . . . . . . . .  19  3.3  The Shapovalov Bilinear Form . . . . . . . . . . . . . . . . . . .  20  3.4  A Positive Definite Inner Product  22  Kac-Moody Algebras: Arithmetic Theory  . . . . . . . . . . . . . .  24  An Integral Form for UC .g/ . . . . . . . . . . . . . . . . . . . .  24  4.1.1  Construction . . . . . . . . . . . . . . . . . . . . . . . .  24  4.1.2  Proof of Theorem 4.4  . . . . . . . . . . . . . . . . . . .  25  . . . . . . . . . . . . . . . . . . . . . . .  27  4.2.1  Construction . . . . . . . . . . . . . . . . . . . . . . . .  27  4.2.2  Proof of Theorem 4.18 . . . . . . . . . . . . . . . . . . .  30  Groups over Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  32  4.1  4.2  5  5.1  6  7  . . . . . . . . . . . . . . . . .  The Chevalley Lattice  HQ .m/  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  33  5.1.1  Definition  33  5.1.2  Hopf Algebra Structure  . . . . . . . . . . . . . . . . . .  35  5.1.3  Examples . . . . . . . . . . . . . . . . . . . . . . . . . .  39  5.2  Peter-Weyl Type Theorems . . . . . . . . . . . . . . . . . . . . .  40  5.3  The Unipotent Subgroup . . . . . . . . . . . . . . . . . . . . . .  42  5.4  The Split Maximal Kac-Moody Group . . . . . . . . . . . . . . .  45  Groups over Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  46  6.1  HZ .m/  6.2  Integrality Conditions  . . . . . . . . . . . . . . . . . . . . . . .  47  6.2.1  Hopf Algebra  . . . . . . . . . . . . . . . . . . . . . . .  47  6.2.2  Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . .  49  6.3  Integral Subalgebras . . . . . . . . . . . . . . . . . . . . . . . .  50  6.4  The Arithmetic Group  . . . . . . . . . . . . . . . . . . . . . . .  52  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  53  Representation Theory . . . . . . . . . . . . . . . . . . . . . . .  53  7.1.1  Constructing the Map  . . . . . . . . . . . . . . . . . . .  53  7.1.2  Subgroups . . . . . . . . . . . . . . . . . . . . . . . . .  55  7.1.3  The Arithmetic Subgroup and the Chevalley Lattice  56  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Structure Theory 7.1  . . .  46  iv  Table of Contents  8  7.2  Existence of a Tits System . . . . . . . . . . . . . . . . . . . . .  57  7.3  Iwasawa Decomposition . . . . . . . . . . . . . . . . . . . . . .  59  7.4  The Orbit of the Highest Weight Vector . . . . . . . . . . . . . .  62  Reduction Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .  64  8.1  The Four Subsets . . . . . . . . . . . . . . . . . . . . . . . . . .  65  8.2  The Arithmetic Set . . . . . . . . . . . . . . . . . . . . . . . . .  66  8.3  Points with Minima . . . . . . . . . . . . . . . . . . . . . . . . .  69  8.3.1  70  Proof of Reduction Theorem . . . . . . . . . . . . . . . € MO . . . . . . . . . . . . . . . . . . . . . . . . . . A Subset of G  72  A Spectral Characterization of I NT.T / . . . . . . . . . .  73  8.4.2 Decay and Minima . . . . . . . . . . . . . . . . . . . . . € [ is not € -invariant . . . . . . . . . . . . . . . . . . . . . . . . G  75 77  8.5.1  Iwasawa Decomposition in SL2 . . . . . . . . . . . . . .  77  8.5.2  The Counterexample . . . . . . . . . . . . . . . . . . . .  78  Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . .  79  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  80  Index of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  83  A Formulas in Associative Algebras . . . . . . . . . . . . . . . . . . .  85  8.4  8.4.1 8.5  8.6  Appendices B Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  87  C Hopf Algebra  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  90  C.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  90  C.2 Enveloping Algebra  . . . . . . . . . . . . . . . . . . . . . . . .  91  C.3 The Dual of the Enveloping Algebra . . . . . . . . . . . . . . . .  91  D Amalgams and Tits Systems . . . . . . . . . . . . . . . . . . . . . .  95  D.1 Direct Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . .  95  D.2 Tits Systems  95  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  v  Table of Contents D.3 Tits’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . .  96  vi  Acknowledgements This thesis would not exist without the advice and encouragement of my advisor, Bill Casselman, who suggested the topic and insisted I finish. I should also mention the generous financial support he has provided me for more than six years, which made this thesis possible in the first place. I am very much indebted to Julia Gordon. Her patience, concern and care went beyond anything required or expected of her. The very least of which was to carefully read an early draft of this thesis and point out many small errors. I would like to thank H. Garland for providing me with original copies of [10] and [13], J. Humphreys for mailing me hard copies of [26] and [27] (which are very hard to find) and S. Kumar for scanning and emailing [23] (another text which is not widely available). While these are not cited in the body of the thesis, reading them was very helpful to me as it enabled me to find the right frame to deal with the problems that are the subject of the thesis.  vii  Chapter 1  Introduction 1.1  The Aim  The goal of this Thesis is to state and prove a reduction theorem for Kac-Moody groups which arise from generalized Cartan matrices (GCMs) which are symmetrizable and invertible. In fact the thesis generalizes the earlier work by H. Garland in [11] and [12] which only deals with untwisted affine GCMs. The only exceptions are §18 and §21 in [12]. The former constructs a fundamental domain for the unipotent subgroup while the latter deals with intersection of the € -orbits with the Siegel set. The question of constructing a fundamental domain does not generalize since his construction relies on the realization of untwisted affine KacMoody algebras as central extensions of loop algebras. As for the intersection of the € -orbits with the Siegel set there is a analogous theory in the general symmetrizable case however this is not covered in the thesis.  1.2  Structure of the Thesis  The thesis has seven chapters beside the current one and each chapter in the thesis begin with short overview of what will follow. The seven chapters can be divided in four parts: Chapters 2 and 3 chapters provide an introduction to theory of Kac-Moody algebras and their representation theory. The material in chapter 4 corresponds to [11], it provides an arithmetic theory for Kac-Moody algebras. In chapters 5, 6 and 7 we define the maximal Kac-Moody group. Most of 1  1.3. Notation the material is from chapter I of [17]. After defining a number of subgroups (including the minimal parabolics and the Borel) O. Mathieu employs techniques from algebraic geometry to define a group by constructing every single Bruhat cell and constructing global product and inverse maps. Instead of doing this we simply use Tits’s work and take the maximal Kac-Moody group to be the product of parabolic subgroups amalgamated along their intersections. Chapter 8 deals with reduction theory of the group constructed earlier and contains all the new results. First we encounter the problem that unlike the finite dimensional case we can not have a reduction theory for the entire group. So one has to find a € -invariant proper subset of the group which is well behaved. There are various candidates, we examine two in detail, prove reduction theory for one of them and show that it contains a large subset. Finally we conclude by listing some open problems.  1.3  Notation  Here is some of the notation used in the thesis: C; N; Q; R; Z: the usual suspects. ˝ always indicates ˝Z . F is a field of characteristic zero. R is a commutative ring with a unit. For any F-vector space VF we let VF denote its linear dual: H OMF.VF; F/. h ; i W VF  VF ! F is the natural pairing between VF, and its dual, VF .  H OMR . ; /: the set of R-module homomorphisms. H OMR  alg .  ; /: the set of R-algebra homomorphisms.  H OMR  lie .  ; /: the set of R-Lie algebra homomorphisms. 2  1.3. Notation UR .l/ the universal enveloping algebra of the R-Lie algebra lR . z R .l/ is the augmentation ideal in UR .l/. U There is also an index of notation which gives the page on which the symbol was first introduced or defined.  3  Chapter 2  Kac-Moody Algebras: A Primer Kac-Moody algebras are a class of Lie algebras generalizing the notion of finite dimensional semi-simple Lie algebras (see Theorem 2.11 below). In this chapter we define the Kac-Moody algebras, and introduce some elementary concepts. Our references for basic theory of Kac-Moody algebras are [7] (chapters 14 16 and 19) and [15].  2.1  Definitions  Notation 2.1. Let n 2 N and set I D f1;  ; ng.  Definition 2.2. A generalized Cartan matrix (or GCM for short) is a square matrix A D Aij i;j 2I satisfying the following:  (1) Aij 2 Z for all i; j 2 I . (2) Ai i D 2 for all i 2 I . (3) Aij Ä 0 if i ¤ j . (4) Aij ¤ 0 if and only if Aj i ¤ 0 for all i; j 2 I . Definition 2.3. Let A be a GCM of size n and corank r. A realization of A is a triplet: .aC ; ˘; ˘ _ /, where: (1) aC is a C-vector space of dimension n C r. (2) ˘ D f˛1 ; ; ˛n g is a linearly independent subset of aC . ˚ « (3) ˘ _ D ˛1_ ; ; ˛n_ is a linearly independent subset of aC . D E (4) ˛i ; ˛j_ D Aj i for all i; j 2 I . 4  2.2. A Classification of GCMs Remark 2.4. If dim.aC / < nCr then one can not find linearly independent subsets ˘  aC and ˘ _  aC which satisfy (4) in Definition 2.3.  Remark 2.5. For a given GCM, a realization always exists and is unique up to isomorphism of vector spaces, see Proposition 14.2 and 14.3 in [7]. Definition 2.6. Let A be a GCM with realization .aC ; ˘; ˘ _ /. The associated Kac-Moody algebra gC is the C-Lie algebra generated by aC and 2n generators fe˙i W i 2 I g, subject to the following relations: ŒaC ; aC  D 0 ei ; e  j  D ıij ˛i_  8i; j 2 I  Œ´; e˙i  D ˙ h˛i ; ´i e˙i  8i 2 I; 8´ 2 aC  And the Serre relations: ad.e˙i /1  Aij  .e˙j / D 0;  8i; j 2 I W i ¤ j:  Remark 2.7. Note that this is not the definition given in [7] and [15], however for the particular class of GCMs we are interested in (symmetrizable Kac-Moody algebras) the two definitions coincide (Theorem 19.30 in [7] or Theorem 9.11 in [15]). Proposition 2.8 ([7] Proposition 14.17). There is an automorphism ! of gC satisfying ! 2 D 1 determined by: !.e˙i / D  2.2  e  i;  !jaC D  1aC :  A Classification of GCMs  Definition 2.9. Two GCMs, A and B are called equivalent if they have the same size n and there is a permutation  of I such that:  Bij D A .i / .j / ;  8i; j 2 I:  5  2.2. A Classification of GCMs Definition 2.10. A GCM, A, is called decomposable if it is equivalent to a diagonal sum A1  0  0  A2  !  of smaller GCMs A1 ; A2 . A GCM that is not decomposable is called indecomposable. If A is a GCM so is its transpose. Moreover A is indecomposable if and only if its transpose is indecomposable. Let v D .v1 ;  ; vn / be a vector in Rn . We write v  0 (v > 0) if vi  0  (vi > 0) for each i . Then one has the following classification of indecomposable GCMs (see [7] Corollary 15.11) into three classes each of which is closed under taking transpose: (1) A has finite type if and only if there exists u > 0 with Au > 0. (2) A has affine type if and only if there exists u > 0 with Au D 0. (3) A has indefinite type if and only if there exists u > 0 with Au < 0. The following justifies our claim that Kac-Moody algebras generalize the notion of finite dimensional simple Lie algebras: Theorem 2.11 ([7] Theorem 15.19). Let A be an indecomposable GCM. Then A has finite type if and only if it is the Cartan matrix of a finite dimensional simple Lie algebra. At any rate from now on we will assume that the GCM is invertible, in particular r D 0. This is not an essential assumption for what we want to do but it will simplify our task.  6  2.3. Root System  2.3  Root System  Definition 2.12. We define two discrete additive subgroup of aC and aC generated by ˘ and ˘ _ : Q D Z˛1 ˚ _  Q D  Z˛1_  ˚  ˚ Z˛n ; ˚ Z˛n_ :  Q is called the root lattice, and Q_ the coroot lattice. Finally set: Q˙ D Z˙ ˛1 ˚ Definition 2.13. For ˛ D P ki .  P  ˚ Z˙ ˛n :  ki ˛i 2 Q the height of ˛ is the number ht.˛/ D  Definition 2.14. Introduce a partial ordering  on aC by setting  2  if  QC . Definition 2.15. For every aC -module VC and every space associated to  2 aC we define the weight  as :  VC; D fv 2 VC W ´ v D h ; ´i v; for all ´ 2 aC g : The elements of VC; are called the weight vectors corresponding to  and the  dimension of VC; is the multiplicity of the weight . The set: ˚  « 2 aC nf0g W VC; ¤ f0g ;  is called the weights of VC . Definition 2.16. With the adjoint action, ´ x D Œ´; x, gC becomes an aC -module. In this particular case the weight vectors and weight spaces are referred to as roots and root spaces respectively, while the weights of gC will be denoted by  and  called the root system of gC . Proposition 2.17 ([7] Proposition 14.18).  7  2.3. Root System (1) gC D  L  ˛2Q gC;˛ .  (2) dim gC;˛ < 1 for all ˛ 2 Q. (3) gC;0 D aC . (4) If ˛ ¤ 0 then gC;˛ D 0 unless ˛ 2 Q. (5) gC;˛ ; gC;ˇ  gC;˛Cˇ for all ˛; ˇ 2 Q. Q. The roots in QC are called  Definition 2.18. Proposition 2.17 shows that positive roots and denoted by denoted by  C  and the roots in Q are called negative roots and  . If we set: M  ˙ nC D  ˛2  gC;˛  ˙  then we have a direct sum of C-vector spaces: C gC D nC ˚ aC ˚ nC ;  This direct sum decomposition is referred to as the triangular decomposition of C gC . The subspaces nC , aC and nC are in fact Lie subalgebras of gC . In accordance  with the finite dimensional case aC is called the Cartan subalgebra of gC . Proposition 2.19 ([7] Proposition 14.19). (1) dim gC;˛i D dim gC;  ˛i  D 1.  (2) If k > 1 then dim gC;k˛i D dim gC;  k˛i  D 0.  Definition 2.20. For each i 2 I we define the following subalgebras: n˙˛i ;C D Ce˙i D gC;˙˛i : C The subalgebra bC D aC ˚ nC is called the Borel algebra of gC . For each i 2 I  the corresponding minimal parabolic algebra is defined as: pi;C D n  ˛i ;C  ˚ bC .  8  2.4. An Analogue of the Killing Form  2.4  An Analogue of the Killing Form  For the finite dimensional semi-simple Lie algebras the Killing form on a finite dimensional semi-simple Lie algebra is defined as: .xjy/ D trace.ad.x/ ı ad.y//: The Killing form has very desirable properties: it is a non-degenerate symmetric bilinear form that is invariant, i.e. one has: .Œx; yj´/ D .xjŒy; ´/. We would like to define an analogue of the Killing form for Kac-Moody algebras that are not of finite type. However since these are infinite dimensional, the expression trace.ad.x/ ı ad.y// is not always defined. Therefore we will impose a further restriction on our GCM so that gC has an analogue of the Killing form. Definition 2.21. A GCM, A, is called symmetrizable if there exists a nonsingular diagonal matrix D D diag.d1 ;  ; dn / and a symmetric matrix B, such that A D  DB.  Remark 2.22. Indecomposable GCMs of finite or affine type are symmetrizable, see Theorem 15.17 in [7]. Definition 2.23. On aC define: ˇ Á ˇ Á ˇ ˇ ˛i_ ˇ˛j_ D ˛j_ ˇ˛i_ D di dj Bij : Since ˘ _ is a basis for aC by extending using linearity we obtain a symmetric bilinear form on the Cartan subalgebra. Proposition 2.24 ([7] Proposition 16.1). The symmetric bilinear form on aC defined above is non-degenerate. Based on Proposition 2.24 we define a bijection aC ! aC given by: ˛ 7! ´˛ where ´˛ is defined by: .´˛ j´/ D h˛; ´i ;  8´ 2 aC ;  in particular we have ˛i_ D di ´˛i . Using this bijection we can define the induced 9  2.5. Integrable Modules bilinear form on aC : ˇ . j / ´ ´ ˇ´  :  ˇ In particular we have: ˛i ˇ˛j D Bij . Theorem 2.25 ([15] Theorem 2.2, Exercise 2.2). If A is symmetrizable then gC has a non-degenerate symmetric bilinear C-valued form such that: (1) . j / is invariant. (2) When restricted on aC , . j / is given by Definition 2.23. ˇ (3) gC;˛ ˇgC;ˇ D 0 unless ˛ C ˇ D 0. (4) Suppose x 2 gC;˛ ; y 2 gC; (5) The pairing gC;˛  gC;  ˛  ˛  then Œx; y D .xjy/ ´˛ .  given by .x; y/ 7! .xjy/ is non-degenerate.  (6) For each 0 ¤ x 2 gC;˛ there exists y 2 gC;  ˛  with Œx; y ¤ 0.  (7) . j / is uniquely determined by (1) and (2). Definition 2.26. The form of Theorem 2.25 is called the standard invariant form on gC . Remark 2.27. When A is of finite type the standard invariant form is a multiple of the Killing form.  2.5  Integrable Modules  Definition 2.28. A linear endomorphism F of vector space VC is called locally nilpotent if for every vector v 2 VC there exists N 2 N such that F N .v/ D 0. Definition 2.29. A representation,  W gC ! gl.VC /, is called integrable if:  VC D  M  VC; ;  2aC  and if .e˙i / are locally nilpotent endomorphisms of VC for all i 2 I . 10  2.5. Integrable Modules Lemma 2.30 ([7] Proposition 7.17). For all i 2 I , ad.e˙i / are locally nilpotent endomorphisms of gC , in other words the adjoint module is integrable. Proof. Since we already know that gC decomposes into aC weight spaces, we only need to show that ad.e˙i / are locally nilpotent linear endomorphisms of gC for all i 2 I . We will show ad.ei / is locally nilpotent, the proof for ad.e i / is similar. First we claim that if ad.ei / acts locally nilpotently on x and y then it also locally nilpotently on Œx; y to see this consider: N X N  N  ad.ei / .Œx; y/ D  kD0  k  ! ad.ei /k .x/; ad.ei /N  ad.ei /k .x/ will be 0 if k is sufficiently large and ad.ei /N k is sufficiently large. Thus ad.ei  /N .Œx; y/  k  k .y/  .y/ ;  will be 0 if N  will be 0 if N is sufficiently large.  Therefore the set of elements of gC on which ad.ei / acts locally nilpotently is a subalgebra. However ad.ei /.ei / D 0 ad.ei /1  Aij  .ej / D 0  i ¤j  ad.ei /2 .˛j_ / D 0 ad.ei /3 .e i / D 0 ad.ei /.e  j/  D0  i ¤j  and therefore the subalgebra contains all the generators, so it is the whole of gC .  11  2.6. Weyl Group  2.6 2.6.1  Weyl Group Definition  Definition 2.31. For a locally nilpotent endomorphism F of a vector space VC we define its exponential, E XP.F /, as the formal sum: E XP.F / D  1 X Fk : kŠ  kD0  Definition 2.32. If  W gC ! gl.VC / is an integrable module for gC then for each  i 2 I we can define: ri D E XP. .ei // E XP. . e i // E XP. .ei // 2 GL.VC /: In particular since the adjoint module is integrable, for each i 2 I we have an automorphism riad 2 GL.gC /. Lemma 2.33 ([7] Propositions 16.11). riad .aC / D aC . For ´ 2 aC we have: riad .´/ D ´  h˛i ; ´i ˛i_ :  Proof. Let ´ 2 aC , we will compute the action of riad term by term. First we have: E XP.ad.ei //.´/ D .1 C ad.ei //.´/ D ´ C Œei ; ´ D ´  h˛i ; ´i ei :  Based on this we add the second term: E XP.ad. e i // E XP.ad.ei //.´/ D E XP.ad. e i // ´ h˛i ; ´i ei à  ad.e i /2 D 1 ad.e i / C ´ h˛i ; ´i ei 2 D ´ h˛i ; ´i ei Œe i ; ´ C h˛i ; ´i Œe i ; ei  C D´  1 2  ad.e i / Œe i ; ´  h˛i ; ´i ei C  1 2  h˛i ; ´i Œe i ; ei   h˛i ; ´i e  ad.e i / h˛i ; ´i e  i i  h˛i ; ´i ˛i_  C h˛i ; ´i ˛i_  12  2.6. Weyl Group D´  h˛i ; ´i ei C  D´  1 2  h˛i ; ´i e  i  h˛i ; ´i ˛i_  0 C h˛i ; ´i .2e i /  h˛i ; ´i ei  h˛i ; ´i ˛i_  Finally: riad .´/ D E XP.ad.ei // E XP.ad. e i // E XP.ad.ei //.´/ D E XP.ad.ei // ´  h˛i ; ´i ei  h˛i ; ´i ˛i_  D .1 C ad.ei // ´  h˛i ; ´i ei  h˛i ; ´i ˛i_  D ´  h˛i ; ´i ˛i_  h˛i ; ´i ei  C Œei ; ´  0  D´  h˛i ; ´i ˛i_  D´  h˛i ; ´i ˛i_  h˛i ; ´i ei ; ˛i_ 2 h˛i ; ´i ei C h˛i ; ´i .2ei /  Definition 2.34. Let ri denote the restriction of riad to aC , then one gets: ri2 D 1aC and ri .˛i_ / D  ˛i_ . In fact we have: ri .´/ D ´  h˛i ; ´i ˛i_ :  ri are called the fundamental reflections, the group they generate (as a subgroup of GL.aC /) is called the Weyl group and is denoted by W . Proposition 2.35 ([7] Proposition 16.13). The bilinear form . j / on aC is W invariant. Proof. Let ´; ´0 2 aC . Then: ˇ ri .´/ˇri .´0 / D ´ ˇ D ´ˇ´0 ˇ D ´ˇ´0 ˇ D ´ˇ´0 ˇ D ´ˇ´0  ˇ h˛i ; ´i ˛i_ ˇ´0  ˝ ˛ ˛i ; ´0 ˛i_ ˇ ˇ ˝ ˛ ˇ ˝ ˛ ˛i ; ´0 ´ˇ˛i_ C ˛i ; ´0 h˛i ; ´i ˛i_ ˇ˛i_ h˛i ; ´i ˛i_ ˇ´0 ˇ ˝ ˛ ˇ ˝ ˛ ˛i ; ´0 ´ˇdi ´˛i C ˛i ; ´0 h˛i ; ´i .2di / h˛i ; ´i di ´˛i ˇ´0 ˝ ˛ ˝ ˛ ˝ ˛ ˛i ; ´0 di h˛i ; ´i C ˛i ; ´0 h˛i ; ´i .2di / h˛i ; ´i di ˛i ; ´0  13  2.6. Weyl Group In fact a converse is true that is if there exists a non-degenerate symmetric W -invariant bilinear form on aC then A is symmetrizable ([15] Exercise 3.3).  2.6.2  Action on the Weight Space  Definition 2.36. We define a W -action on aC , let w 2 W and  2 aC then the  weight w. / is defined as follows: 8´ 2 aC W  hw. /; ´i D  ˝  ;w  1  ˛ .´/ :  Lemma 2.37. The W -action on aC is compatible with the bijection ˛ 7! ´˛ and hence the induced bilinear form . j / on aC is W -invariant as well. Proof. Let w. / D  for ;  2 aC and take ´ 2 aC to be arbitrary:  ˇ .w.´ /j´/ D ´ ˇw  1  ˝  1  ˛ .´/ ˇ ˇ D hw. /; ´i D h ; ´i D ´ ˇ´ D ´w. / ˇ´ .´/ D  ;w  Since . j / is non-degenerate and ´ 2 aC is arbitrary we have: w.´ / D ´w. / . 2 aC and observe:  For the W -invariance, let ; ˇ ˇ rj . /ˇrj . / D ´rj . / ˇ´rj .  /  ˇ ˇ D rj .´ /ˇrj .´ / D ´ ˇ´  D . j /:  Lemma 2.38 ([7] Proposition 16.14). The action of ri on aC is given by: ri . / D ˝ ˛ ; ˛i_ ˛i . Proof. Let  2 aC ; ´ 2 aC : hri . /; ´i D h ; ri .´/i ˝ ˛ D ; ´ h˛i ; ´i ˛i_ ˝ ˛ D h ; ´i ; ˛i_ h˛i ; ´i ˝ ˝ ˛ ˛ D ; ˛i_ ˛i ; ´  Proposition 2.39 ([7] Proposition 16.15). If ˛ 2  ; w 2 W then w.˛/ 2  .  Moreover dim gC;˛ D dim gC;w.˛/ . 14  2.7. Geometry of the Weyl Group  2.6.3  W as a Coxeter Group  Theorem 2.40 ([7] Theorem 16.17). The Weyl group W is a Coxeter group generated by r1 ;  ; rn with relations: ri2 D 1 2  ri rj  3  ri rj  4  ri rj  6  ri rj  2.7 2.7.1  D1  if Aij Aj i D 0  D1  if Aij Aj i D 1  D1  if Aij Aj i D 2  D1  if Aij Aj i D 3  Geometry of the Weyl Group Real and Imaginary Roots  Definition 2.41. ˛ 2  is called a real root if there exist ˛i 2 ˘ and w 2 W such  that ˛ D w.˛i /, the set of all real roots is denoted by  re .  A root that is not real  is called imaginary and the collection of all imaginary roots is indicated by Finally  re ˙;  im ˙  im .  are defined as the 4 possible intersections of real and imaginary  roots with positive and negative ones. Remark 2.42. Real roots behave very much like the roots of finite dimensional semi-simple Lie algebras: they have multiplicity 1 and the only multiples of a real root ˛ that themselves are roots are ˙˛ ([7] Proposition 16.18). Imaginary roots on the other hand have no counterpart in the finite dimensional Lie algebras (see Proposition 16.27 in [7]). Moreover if ˛ 2  im C  then k˛ 2  im C  for all k 2 ZC  ([7] Corollary 16.25) which in turn implies that gC is infinite dimensional exactly when A is not of finite type.  2.7.2  The Tits Cone  Definition 2.43. Given a GCM A of size n let aR be an n-dimensional R-vector space such that C ˝R aR ; ˘; ˘ _ is a realization for A. In our case, since we  15  2.7. Geometry of the Weyl Group assume A to be invertible, we may take aR to be the R-subspace in aC generated by simple coroots. Definition 2.44. We define the Tits cone and the open Tits cone as the following subsets in aR : ˚ T D ´ 2 aR W h ; ´i < 0; for finitely many ˚ I NT.T / D ´ 2 aR W h ; ´i Ä 0; for finitely many  2  re C  «  2  re C  «  Here I NT.T / is the interior of T in the metric topology of aR . Remark 2.45. A GCM A has finite type if and only if T D I NT.T / D aR . The subsets T ; I NT.T / are closely related to the action of the Weyl group: Definition 2.46. For each subset J  I the corresponding face in aR is defined as  follows: FJ D f´ 2 aR W h˛i ; ´i D 0; 8i 2 J and h˛i ; ´i > 0; 8i … J g : Given any subset J  I we say J has finite type if the principal submatrix corre-  sponding to J has finite type. Now define: DD  [ J  FJ  I  [  Dfin D  FJ  J I J has finite type  Proposition 2.47 ([2] Proposition 4.4.9). (1) T D W D. (2) I NT.T / D W Dfin . The following characterization of the closure of the Tits cone in the metric topology of aR will be useful later:  16  2.7. Geometry of the Weyl Group Proposition 2.48 ([29] Proposition 5.6). If A is of indefinite type then: ˚ C L.T / D ´ 2 aR W h ; ´i  0; 8 2  im C  «  :  17  Chapter 3  Kac-Moody Algebras: Highest Weight Modules In this chapter we introduce the concept of a highest weight module. Almost all of the theory of this class of representations is similar to that of the finite dimensional case; one major difference is that when our GCM is not of finite type then the irreducible quotient of the Verma module is not finite dimensional. One of the most important aspects of the theory is the Shapovalov bilinear form (see Definition 3.11), originally introduced in [22]. This provides us with a non-degenerate symmetric contravariant bilinear form on any irreducible highest weight module (see Proposition 3.13). Furthermore, if the irreducible highest weight module is integrable as well, we get a positive definite inner product.  3.1  Verma Modules  Definition 3.1. Let by  C nC  2 aC and define KC to be the left ideal of UC .g/ generated  and all elements of the form ´  with highest weight  h ; ´i where ´ 2 aC . The Verma module  is defined as: M. /C D UC .g/=KC :  Proposition 3.2 ([7] §19.1). Let 1 2 M. /C be the image of 1 2 UC .g/. Then: (1) Every element of M. /C is uniquely expressible in the form of u 1  for  some u 2 UC .n /. L (2) M. /C D 2a M. /C; . C  18  3.2. The Irreducible Quotient (3) M. /C; ¤ 0 if and only if  .  Another way of defining the Verma module is as follows: let C  be a 1-  dimensional vector space on which aC acts via h ; i. We extend this to a repC C resentation of bC D aC ˚ nC by requiring that nC act trivially. Now we have: g  M. /C ´ I NDbC C C  D UC .g/ ˝UC .b/ C :  Lemma 3.3. The product map gives us an isomorphism of C-vector spaces: UC .n / ˝C UC .a/ ˝C UC .nC / Š UC .g/: Proof. This follows from the triangular decomposition of gC and the PBW Theorem. In particular we have: UC .g/ Š UC .n / ˝C UC .b/. Therefore as UC .n /modules, M. /C Š UC .n /, where UC .n / acts on itself via left multiplication. In other words as UC .n /-modules, all Verma module look the same.  3.2  The Irreducible Quotient  Definition 3.4. The Verma module has a unique maximal proper submodule ([7] Theorem 10.9) which we shall denote by M 0 . /C . We define: L. /C D M. /C =M 0 . /C : Then L. /C is an irreducible module and therefore it is called the irreducible highest module with highest weight  .  Remark 3.5. From Proposition 3.2 we see that L. /C D  M  L. /C;  2aC  and that every weight  that appears in L. /C has to be of the form  ˛, where  ˛ 2 QC . 19  3.3. The Shapovalov Bilinear Form Definition 3.6. The set of weights of L. /C will be denoted by P . The depth of ˛ 2 P , denoted by dp. /, is taken to be the ht.˛/. ˛ ˝ Definition 3.7. 2 aC is called integral if ; ˛i_ 2 Z for all i 2 I . It is called ˛ ˝ dominant if ; ˛i_ > 0 for all i 2 I . the weight  D  Proposition 3.8 ([7] Proposition 19.14; [16] Corollary 2.1.8). L. /C is integrable if and only if  3.3  is dominant and integral.  The Shapovalov Bilinear Form  Definition 3.9. Consider the involution ! W gC ! gC we have from Proposition 2.8 and let U.!/ W UC .g/ ! UC .g/ be its lift to the universal enveloping algebra. Now define:  D U.!/ ı , where  is the principal anti-automorphism  of UC .g/ (see Appendix C). Definition 3.10. Based on Lemma 3.3 we can write UC .g/ as a direct sum of two vector spaces: C UC .g/ D UC .a/ ˚ nC UC .g/ C UC .g/ nC :  Let Á denote the projection on the first factor, this map is commonly referred to as the Harish-Chandra map. Definition 3.11. The Shapovalov bilinear form is defined as follows:  ˚  S W UC .g/  UC .g/ ! UC .a/  S.x; y/ ´ Á. .x/y/  Proposition 3.12 ([18] §2.8 Proposition 1). (1) S is symmetric. (2) For all x; y; u 2 UC .g/; S.ux; y/ D S.x;  .u/y/.  (3) For ˛ ¤ ˇ 2 Q; UC .g/˛ ?UC .g/ˇ . 20  3.3. The Shapovalov Bilinear Form (4) S.1; 1/ D 1. Since ˘ _ is a basis for the abelian Lie algebra aC (recall our assumption that 2 aC can be extended to the polynomial  the GCM A is of full rank), any weight ring: UC .a/ Š C  ˛1_ ;  ; ˛n_  in a natural way:  ˛i_ ˛j_ D  ˝  ; ˛i_  ˛D  E ; ˛j_ :  Now we define a bilinear form: S W M. /C  M. /C ! C;  as follows. Let v; w 2 M. /C then by Proposition 3.2 there exist x; y 2 UC .n / such that v D x 1 ; w D y 1 and set: S .v; w/ D  .S.x; y//:  Proposition 3.13 ([16] Proposition 2.3.2). (1) S is symmetric. (2) S is contravariant, that is: S .x v; w/ D S .v;  .x/ w/;  for all v; w 2 M. /C and all x 2 UC .g/. (3) S .M. /C; ; M. /C; / D 0 if  ¤ .  (4) S .M 0 . /C ; M. /C / D 0. (5) S induces a non-degenerate symmetric contravariant bilinear form on L. /C also denoted by S . (6) Any contravariant bilinear form on L. /C is a scalar multiple of S  and  hence it is automatically symmetric.  21  3.4. A Positive Definite Inner Product Remark 3.14 ([15] §9.4). For any highest weight vector v 2 L. /C; , we define a corresponding functional: Ev Œ  W L. /C ! C as follows: v D Ev Œvv C v 0 ;  v0 2  M  L. /C; :  ¤  Then one may write: S .x v ; y v / D Ev Œ .x/y v : A normalization such as S .1 ; 1 / D 1 will determine the bilinear form uniquely, we will use this normalization from now and we will abbreviate E1 Œ  to EŒ .  3.4  A Positive Definite Inner Product  Definition 3.15. Let gR be the real subalgebra of gC generated by fe˙i W i 2 I g and aR . This gives us a conjugate linear involution of gC denoted by u 7! u which we lift to UC .g/. Since the involution ! satisfies: !.gR /  gR and we have: !.u/ D !.u/ we can define a conjugate linear anti-automorphism of UC .g/ of order two by setting For any  0  D  ( was introduced in Definition 3.9).  2 aR we get a real form: L. /R ´ gR 1  L. /C and hence a  conjugate linear involution of L. /C , denoted by v 7! v. We define a Hermitian form f ; g on L. /C by: fv; wg D S .v; w/: Since S was a contravariant bilinear form, f ; g becomes a contravariant Hermitian form, this means that we have: fx v; wg D fv;  0 .x/  wg :  In order to get an inner product on L. /C we need f ; g to be positive definite. Using the contravariance of f ; g we can calculate some inner products. For exam-  22  3.4. A Positive Definite Inner Product ple: fe  i  1 ;e  i  1 g D f1 ;  0 .e i /e i  D f1 ; ei e  i  1 g  1 g  D f1 ; .Œei ; e i  C e i ei / 1 g ˚ « D 1 ; ˛i_ 1 ˚ ˝ ˛ « D 1 ; ; ˛i_ 1 ˝ ˛ D ; ˛i_ f1 ; 1 g ˝ ˛ If we use a normalization such as f1 ; 1 g D 1 we see that ; ˛i_ 0 is a necessary condition for f ; g being positive definite. In fact with similar calculations one can show that  being dominant and integral is necessary. However it turns  out that this condition is sufficient as well: Theorem 3.16 ([16] Theorem 2.3.13). f ; g is positive definite on L. /C if and only if is dominant and integral. p Notation 3.17. For v 2 L. /C we set: kvk D fv; vg.  23  Chapter 4  Kac-Moody Algebras: Arithmetic Theory In any integrable highest weight module, L. /C , we would like to define a lattice which is compatible with the inner product defined on L. /C . That is, the inner product of any two elements in the lattice is an integer, in particular the length of any element is a positive integer. How should we define such a lattice? Recall that the highest weight vector generates the module: L. /C D UC .g/ 1 . Based on this we may define L. /Z D UZ .g/ 1 where UZ .g/ itself is a lattice in UC .g/ and we show that this is the lattice in L. /C with the desired properties. This chapter is divided in two sections: in §1 we first define what we mean by a lattice in UC .g/ (see Definition 4.1) and then construct one by giving generators. In §2 we show that the subset defined in the highest weight module is a lattice with all the desired properties. The material of §1 is based on [28] §4.4 while §2 follows [11], we note that while the [11] only deals with the specific case of affine GCMs the same proof can be used for the general case as we show here.  4.1 4.1.1  An Integral Form for UC .g/ Construction  Definition 4.1. An integral form of a C-algebra AC is a subring A  AC such that  the canonical map: C ˝ A ! AC is bijective. 0 Notation 4.2. In order to simplify our notation in this section we will use UC ; UC ˙ and UC as a shorthand for UC .g/; UC .a/ and UC .n˙ / respectively.  24  4.1. An Integral Form for UC .g/ Definition 4.3. We define the following subrings of UC (for notation see Appendix A): U˙i D  M  Œp  Ze˙i  p2N  * 0  U D  ´  !  p  + _  W ´ 2 Q ;p 2 N  ˝ ˛ U˙ D U˙1 ; ; U˙n ˝ ˛ U D U ; U0 ; UC Theorem 4.4 ([28] §4.4). U is an integral form for UC . The proof of Theorem 4.4 can be divided in two steps: C 0 (1) U ; U0 and UC are integral forms for UC ; UC and UC respectively.  (2) The product map: U ˝ U0 ˝ UC ! U is bijective. (1) and (2) combined with Lemma 3.3 would imply that U is an integral form for UC .  4.1.2  Proof of Theorem 4.4  Proof of (1) 0 Proposition 4.5. U0 is an integral form for UC . 0 Proof. UC Š C ˛1_ ;  ; ˛n_ and ˛1_ ;  ; ˛n_ 2 U0 .  ˙ Proposition 4.6. U˙ is an integral form for UC . C Proof. Since UC is a subring of UC , we only need to show that the canonical map C C ˝ UC ! UC is bijective. It is surjective because UC contains all the generators C of UC . Assume it is not injective, that is:  0¤  k X i D1  ci ˝ xi 7!  k X  ci xi D 0;  i D1  25  4.1. An Integral Form for UC .g/ where xi are monomials in UC and ci 2 C are all nonzero. Using the grading C of UC we see that all xi have the same degree m. Therefore fx1 ;  ; xk g  C UC  is linearly independent over Z but not over C. This contradiction proves that the C canonical map is injective as well and hence UC is an integral form for UC . The  proof for U is identical. Proof of (2) Lemma 4.7. For ´ 2 aC and p; q 2 N we have: ´ p  ! Œq e˙i  D  ! ´ ˙ q h˛i ; ´i : p  Œq e˙i  Proof. Note that Œ´; e˙i  D ˙ h˛i ; ´i e˙i and then use Lemma A.3 with P .X / D X p  .  Lemma 4.8. U0 U˙i D U˙i U0 . Proof. This follows from Lemma 4.7. Lemma 4.9. Ui U0 U  i  D U i U0 Ui .  Proof. Use Lemma 4.8 to arrive at: Ui U0 U  i  Using Lemma A.6 with x D ei ; y D e  D Ui U i U0 i  Œp Œq i  and ´ D ˛i_ we see that: ei e  written as a sum, where each summand belongs to U  i U0 Ui .  can be  Hence:  Ui U i U0 D U i U0 Ui U0 Using Lemma 4.8 one more time gives us the result. Lemma 4.10. If j; i1 ;  ; im is a sequence of m C 1 elements in I then: Ui1 Ui2  U im U  j  U  j  U0 UC :  Proof. We prove this by induction on m: 26  4.2. The Chevalley Lattice m D 1: If j D i then Ui U  i  m D 1 and j ¤ i then Ui U that ei and e  j  U i U0 UC follows from Lemma 4.9. If j  U  j U0 UC  is a consequence of the fact  commute.  m > 1: U i1 U i2  U im U  j  D U i1 U i2  U im U j Á U i1 U j U 0 U C Á D U i1 U j U 0 U C Á U j U0 UC U0 UC DU  j  U0 UC U0 UC  DU  j  U0 U0 UC UC  U  j  U0 UC  Á induction hypothesis  base of induction  Lemma 4.8  Proposition 4.11. The product map U ˝ U0 ˝ UC ! U is bijective. Proof. Lemma 3.3 implies the injectivity of the product map. In order to prove surjectivity let U0 denote the the image of the product map. Then: U0 U  i  U0  0  i  0  UU  U0 U0 So U0 U  4.2 4.2.1  U  U0  Lemma 4.10 Ui  UC  Lemma 4.8  U0 which implies that U0 contains U.  The Chevalley Lattice Construction  Definition 4.12. Set: gZ ´ UZ .g/ \ gC  27  4.2. The Chevalley Lattice ˙ ˙ nZ ´ UZ .n˙ / \ nC  aZ ´ UZ .a/ \ aC D Q_ In particular this allows us to define all these Lie algebras over any commutative ring of characteristic zero with a unit. Definition 4.13. Let  be an integral and dominant weight and define the Cheval-  ley lattice, as L. /Z ´ UZ .g/ 1  L. /C . Then Chevalley lattice is a UZ .g/-  invariant Z-module in L. /C , below we will show that it is indeed a lattice in L. /C (see Theorem 4.18). Lemma 4.14 ([11] Lemma 11.4). EŒ  takes integer values on L. /Z . Proof. Let v 2 L. /Z , by definition there exists a 2 UZ .g/ such that v D a 1 . Define: z Z .n˙ / ´ U z Q .n˙ / \ UZ .n˙ /; U which is the integral span of all monomials in UZ .n˙ / of strictly positive degree. Then we have: z Z .nC / C UZ .a/ z Z .n /UZ .g/CUZ .g/U UZ .g/ D U z Z .nC / 1 D 0 U M z Z .n /UZ .g/ 1 U L. /C; ¤  Hence there exists a0 2 UZ .a/ such that: EŒv D EŒa 1  D EŒa0 1  :  But by definition a0 2 UÁ Z .a/ is an integral linear combination of products of _ ˛ elements of the form: mi which act on v as follows: ˛i_ m Now since  !  ˝ 1 D  is dominant and integral  ˝  ; ˛i_ m  ˛! 1 :  ˛ ; ˛i_ 2 ZC for all i 2 I . 28  4.2. The Chevalley Lattice Theorem 4.15. If v; w 2 L. /Z then fv; wg 2 Z. Proof. By definition there exist a; b 2 UZ .g/ such that v D a 1 ; w D b 1 . Hence: fv; wg D S .v; w/ D S .v; w/ D  .S.a; b// D  Á. .a/b/  where Á is the Harish Chandra map. Since UZ .g/ is -invariant, .a/b 2 UZ .g/. Moreover we may define a map ÁZ W UZ .g/ ! UZ .a/ such that the following diagram commutes: UZ .g/ ÁZ    UZ .a/  / UC .g/   Á  / UC .a/  Therefore we have: fv; wg D But  Á. .a/b/ D  ÁZ . .a/b/ :  is dominant and integral so when extended to UZ .a/ it will only produce  integer values. Corollary 4.16. If v 2 L. /Z then kvk > 1. Definition 4.17. An admissible basis for L. /C is an ordered basis consisting of weight vectors ordered such that the depth of basis elements is non-decreasing in this basis. That is if fv1 ; v2 ; i < j implies dp. i / Ä dp.  g is a an admissible basis with vk 2 L. /C; j /.  k  then  Note that the first basis element of any admissible  basis has to be a highest weight vector, that is it belongs to L. /C; . Theorem 4.18 ([11] Theorem 11.3). L. /C has an admissible basis such that its Z-span is UZ .g/-invariant.  29  4.2. The Chevalley Lattice  4.2.2  Proof of Theorem 4.18  Set: L. /Z; D L. /Z \L. /C; . Then L. /Z has a direct sum decomposition: M  L. /Z D  L. /Z; :  2P  Let B be a Z-basis for L. /Z; and set: BD  [  B ;  2P  we claim B is the basis we are looking for. There are 3 points to prove: (1) The Z-span of B is UZ .g/-invariant. (2) B spans L. /C . (3) B is linearly independent over C. Since L. /Z is the Z-span of B, (1) is true. (2) follows from L. /C D UC .g/ 1 and the fact that UZ .g/ is a lattice in UC .g/. That leaves (3), however this is equivalent to to proving that for any finite subset of L. /Z linear independence over Z implies linear independence over C. We prove the latter by contradiction, so suppose there exist v1 ;  ; vr 2 L. /Z which are linearly independent over Z,  but not over C, that there exist c1 ; r X  ; cr 2 C not all zero such that: cj vj D 0:  (4.19)  j D1  Moreover we assume that r is minimal, in other words any other subset of L. /Z of size smaller than r is not a counterexample to our claim. In order to derive a contradiction we first need the following Lemma: Lemma 4.20. Let v1 ;  ; vr 2 L. /Z be such that v1 ¤ 0 and  with cj 2 C. Then there exist integers n1 ; r X  Pr  j D1 cj vj  D0  ; nr with n1 ¤ 0 satisfying:  cj nj D 0:  j D1  30  4.2. The Chevalley Lattice Proof. Choose a 2 UZ .g/ such that EŒa v1  ¤ 0. Such an element exists, otherwise v1 would generate a non-trivial submodule of L. /C which did not intersect L. /C; , and by definition of L. /C , this is impossible. Applying P P first a and then EŒ  to jr D1 cj vj D 0, we get: jr D1 cj E a vj D 0. Since vj 2 L. /Z ; a 2 UZ .g/ we have E a vj 2 Z from Corollary 4.14. Now take nj ´ E a vj , note that n1 ¤ 0 due to our choice of a. Applying Lemma 4.20 to (4.19) we see that there exist integers n1 ; with n1 ¤ 0 satisfying:  r X  cj nj D 0:  ; nr 2 Z  (4.21)  j D1  Now from (4.19) and (4.21) we have: 0D  r X  ! cj vj n1 D c1 n1 v1 C  j D1  0D  r X  r X  cj n1 vj  j D2  ! cj nj v1 D c1 n1 v1 C  j D1  r X  cj nj v1  j D2  Eliminating c1 n1 v1 using the two equations we get: r X  cj n1 vj  nj v1 D 0  j D2  Set wj WD n1 vj since v1 ;  nj v1 . Now w2 ;  ; wr 2 L. /Z are linearly dependent over Z  ; vr were linearly independent over Z however we have: r X  cj wj D 0:  j D2  which implies w2 ;  ; wr is linearly dependent over C which contradicts the min-  imality of r.  31  Chapter 5  Groups over Q In this chapter we give a definition of the split maximal Kac-Moody group over Q (and any field of characteristic zero) associated to a given GCM. One should note that given a GCM that is not of finite type one can associate at least two different groups with it (maximal vs minimal). Moreover in either case there are several definitions in the literature (see [20] for various definitions of both the maximal and minimal groups and their comparison). The definition given below is new but it is a synthesis of two main ways of approaching the subject. Our starting point is the result that any complex semi-simple Lie group can be expressed as the amalgamated product of the minimal parabolic subgroups and the normalizer of a maximal torus (see Theorem D.2). We use this theorem as our definition: first define the minimal parabolic subgroups and the normalizer of the maximal torus and then we define the split maximal Kac-Moody group as their amalgamated product. The first step (defining the subgroups) is the subject of §1, for any subalgebra mQ  gQ which is invariant under the adjoint action of aQ we construct a Q-  algebra: HQ .m/. Then we show that HQ .m/ is in fact a Q-Hopf algebra and so we have an affine group scheme over Q. In §2 we compute the Hopf algebra HQ .m/ in C terms of the representations of mQ where mQ D aQ ; nQ ; bQ ; li;Q ; pi;Q (for these  two sections we follow Mathieu, pages 19-20 and 24-25 in [17]). In §3 we give an C explicit description of the unipotent group, NQ , as a subset of a completion of the C universal enveloping algebra of nQ . Finally in §4 we note that while gQ itself is  invariant under the adjoint action of the torus, when the GCM is not of finite type gQ becomes infinite dimensional and while this process yields us a group, it is too small to be of any use, see Remark 5.32. The second step (amalgamation) was the approach championed by Jacques Tits, an exposition can be found in [16]. Finally another way of constructing Kac-Moody groups is to use integrable 32  5.1. HQ .m/ representations of the Kac-Moody algebra, see [6].  5.1 5.1.1  HQ .m/ Definition  Notation 5.1. For a subalgebra mQ  gQ , set: ˙ ˙ mQ ´ mQ \ nQ 0 mQ ´ mQ \ aQ :  Definition 5.2. The character lattice is defined as follows: P ´ H OMZ .aZ ; Z/: P is a lattice in aQ which contains the root lattice, Q. For each i 2 I we also set the following subset of the character lattice: Pi ´  ˚  2PW  ˝  ; ˛i_  ˛  « 0 :  Let mQ  gQ be a subalgebra such that: aQ ; mQ  mQ :  (?)  Let L.u/; R.u/ W UQ .m/ ! UQ .m/ be the left and right multiplications by u. Corresponding to these maps we have the left regular representation of UQ .m/ W u 7! L.u/ and the right regular representation: u 7! R. .u//, where  is the  principal anti-automorphism of UQ .m/. The transpose of these maps give us an action of UQ .m/ on its dual: L .u/. / .x/ D .ux/ R .u/. / .x/ D .x .u//  33  5.1. HQ .m/ For ´ 2 aQ let ad.´/ W UQ .m/ ! UQ .m/ denote the lift of ad.´/ W mQ ! mQ . Again the transpose gives us an action of aQ on UQ .m/: ad .´/. / .x/ D .ad.´/.x//: 2 UQ .m/ is called L -finite (resp. R -finite) if the span of the  Definition 5.3.  maps L .u/. / (resp. R .u/. /), as u varies over UQ .m/, is finite dimensional in UQ .m/ . L R Definition 5.4. Let HQ .m/ (resp. HQ .m/) be the set of all linear combinations of  elements  2 UQ .m/ that satisfy:  is L -finite (resp. R -finite).  (1)  (2) There exists  2 Q such that ad .´/. / D h ; ´i  for all ´ 2 aQ .  2 P such that L .´/. / D h ; ´i 0 . h ; ´i ) for all ´ 2 mQ  (3) There exists  (resp. R .´/. / D  L R Lemma 5.5. HQ .m/ D HQ .m/.  Proof. Let . ; ; / be a triplet satisfying the conditions of Definition 5.4 with non-zero. By definition showing  R L R 2 HQ .m/ will imply HQ .m/  HQ .m/.  Set: ˚ « IQ ´ u 2 UQ .m/ W L .u/. / D .u / D 0 : Since  (5.6)  is L -finite, IQ is a right ideal of finite co-dimension in UQ .m/. Therefore  IQ contains, JQ , the annihilator of UQ .m/=IQ , which is a two-sided ideal of of finite co-dimension (it is the largest two sided ideal contained in IQ ). Now consider the following: R  .JQ / . / .x/ D .xJQ / D .JQ / D 0:  Since JQ is invariant under Next let ´ 2  0 mQ ,  and has finite co-dimension  is R -finite.  by definition: ad .´/. / D h ; ´i ; L .´/. / D h ; ´i : 34  5.1. HQ .m/ Therefore: R .´/. / D L .´/. / since  2 P;  ad .´/. / D h  2 Q  P we have  ; ´i ;  L R 2 P. The proof of HQ .m/ à HQ .m/ is  similar. L R Notation 5.7. Based on Lemma 5.5 we set: HQ .m/ ´ HQ .m/ D HQ .m/.  Example 5.8. For mQ D aQ the three conditions enumerated in the definition of HQ .a/ collapse to one. So HQ .a/ is the linear combination of elements  2  UQ .a/ that satisfy: 9 2PW Since UQ .a/ Š Q ˛1_ ;  L .´/. / D h ; ´i :  ; ˛n_ we see that  is determined by  up to a scalar  constant, that is .´/ D h ; ´i .1/. Now the map: 7! .1/ı ; where ı W P ! Q is the map that sends  to 1 and is zero on the rest of the lattice,  shows that HQ .a/ D QŒP, where QŒP is the group algebra of the discrete group P.  5.1.2  Hopf Algebra Structure  Definition 5.9. Let XQ .m/ be the set of all left ideals, IQ , in UQ .m/ that satisfy: (1) IQ is of finite co-dimension in UQ .m/. (2) IQ is stable under the adjoint action of aQ . (3) There exists a finite subset  0  P such that their restriction to mQ satisfies:  0 8´ 2 mQ W  Y  ´  h ; ´i 2 IQ :  2  Lemma 5.10. For  2 UQ .m/ the following are equivalent: 35  5.1. HQ .m/ (1)  2 HQ .m/.  (2) There exists a two-sided ideal JQ 2 XQ .m/ such that .JQ / D 0. Proof. First note that if IQ 2 XQ .m/ then UQ .m/=IQ is a finite dimensional QW UQ .m/ ! Q such that  vector space and any linear map  .IQ / D 0 belongs  to HQ .m/. For the converse, let  2 HQ .m/ be arbitrary, then we may write it as a linear  combination of non-zero elements: D c1  1  C  C cm  where for each k; 1 Ä k Ä m, the triplet .  k;  m;  k;  Definition 5.4. In the proof of Lemma 5.5 for each  k/ k  satisfies the conditions of  we have defined a right ideal  Ik;Q and a two sided ideal Jk;Q contained in it. Now set: JQ D J1;Q \  \ Jm;Q :  Clearly JQ is a two sided ideal of finite co-dimension and .JQ / D 0. Since JQ is a two sided ideal it is stable under the adjoint action in particular by elements of aQ . Finally for condition (3) of Definition 5.9 we take: Df  1;  ;  mg :  Lemma 5.11. HQ .m/ is a subalgebra of the commutative algebra UQ .m/ . Proof. Recall that UQ .m/ is a commutative algebra with the unit map product map  (see Appendix C). Since HQ .m/ is a Q-subspace of UQ .m/ we  only need to show that it is closed under  . In other words:  HQ .m/ ˝Q HQ .m/ 2 HQ .m/ and let  Suppose ;  denote  HQ .m/: . ˝  /. By Lemma 5.10  HQ .m/ is equivalent to finding a two-sided ideal PQ 2 XQ .m/ such that . 0. Since  ;  and  2 /.PQ / D  2 HQ .m/ there exist two-sided ideals IQ ; JQ 2 XQ .m/ such 36  5.1. HQ .m/ .IQ / D  that .  .JQ / D 0. Let PQ D IQ JQ , then one immediately has:  /.PQ / D 0, all that is left to show is PQ 2 XQ .m/. PQ has finite co-dimension: take fx1 ;  ; xp g 2 UQ .m/ to be a basis for the  Q-vector space UQ .m/=JQ and let fy1 ;  ; yq g be a set that generates IQ as an  ideal. Then t 2 IQ can be written as: tD  q X  ui yi ;  (5.12)  i D1  where ui 2 UQ .m/. For each ui we can write: ui D  p X  ! cj vj  C JQ ;  (5.13)  j D1  where cj 2 Q. Combining (5.12) and (5.13) we arrive at the following: tD  D  D  q X  p X  i D1  j D1 p X  q X  iD1 j D1 p q X X  ! cj vj  ! C JQ yi  !  cj vj yi  C JQ yi !  cj vj yi  C IQ JQ  i D1 j D1  which shows that IQ =IQ JQ is finite dimensional and since: dim UQ .m/=IQ JQ D dim UQ .m/=IQ C dim IQ =IQ JQ ; we see that PQ D IQ JQ has finite co-dimension. PQ is stable under the adjoint action of aQ : let a 2 IQ ; b 2 JQ and ´ 2 aQ then: Œ´; ab D ´ab D ´ab  ab´ a´b C a´b  ab´  37  5.1. HQ .m/ D Œ´; ab C aŒ´; b: Since IQ ; JQ both are stable under the adjoint action of aQ we have: Œ´; a 2 IQ ; Œ´; b 2 JQ . Finally from IQ ; JQ 2 XQ .m/ we have two finite sets: the union  1  [  2  1;  2   P. For PQ  satisfies the desired conditions.  Theorem 5.14. If the subalgebra mQ  gQ satisfies .?/ then HQ .m/;  ;  ;  ;  ;  is a commutative Hopf algebra over Q. Proof. Based on Lemma 5.11 and the fact that UQ .m/ is itself a cocommutative Hopf algebra we only need to prove that HQ .m/ is closed under the transpose ;  maps:  and  . The first two are easy to verify and so turn our attention to  the transpose of the product map in UQ .m/:  ˚  W UQ .m/ ! UQ .m/ ˝Q UQ .m/ . /.u ˝ u0 / D  . .u ˝ u0 // D  .uu0 /  2 HQ .m/ and so by Lemma 5.10 there exists JQ 2 XQ .m/  On the other hand  such that .JQ / D 0. Therefore we get: . / JQ ˝Q UQ .m/ C UQ .m/ ˝Q JQ D 0: Combining this with: . / 2 H OMQ UQ .m/ ˝Q UQ .m/; Q ; we get:  . / 2 H OMQ  à UQ .m/ ˝Q UQ .m/ ;Q JQ ˝Q UQ .m/ C UQ .m/ ˝Q JQ  D H OMQ UQ .m/=JQ ˝Q UQ .m/=JQ ; Q D H OMQ UQ .m/=JQ ; Q ˝Q H OMQ UQ .m/=JQ ; Q 38  5.1. HQ .m/ But since JQ 2 XQ .m/ we have: H OMQ UQ .m/=JQ ; Q  HQ .m/: Therefore: . / 2 HQ .m/ ˝Q HQ .m/: Definition 5.15. Based on Theorem 5.14 for any subalgebra mQ  gQ that satisfies .?/ we may define an affine group scheme over Q: MQ D H OMQ  5.1.3  alg .HQ .m/;  Q/:  Examples  Example 5.16. Let us return to the case when mQ D aQ , we have already shown that HQ .a/ D QŒP. Therefore: AQ D H OMQ  alg .QŒP;  Q/ D H OMZ .P; Q /:  So our definition agrees with that of a classical finite dimensional split torus. Example 5.17. For each i 2 I define: li;Q D n  ˛i ;Q  ˚ aQ ˚ n˛i ;Q :  Then li;Q satisfies .?/ and so we get a group Li;Q , this is a finite dimensional reductive group of semi-simple rank 1 which contains AQ as a subgroup. Let W i;Q denote the normalizer of AQ in Li;Q . Then the quotient W i;Q =AQ is a group with two elements: f1; r i g and the conjugation action of r i on AQ is induced from the action of the fundamental reflection ri on aQ . Let a D E XP.´/ 2 AQ then: ri .a/ D r i ar i  1  D ri .E XP.´// D E XP.ri .´//:  39  5.2. Peter-Weyl Type Theorems Moreover we have r 2i 2 AQ , more precisely:  ˚  r 2i W P ! Q r 2i . / D . 1/h  ; ˛i_ i  ˙ Remark 5.18. Earlier we have defined several subalgebras of gQ W aQ ; bQ ; nQ ; n˙˛i ;Q  and pi;Q . Now for each i 2 I we define: M  ni;Q D  ˛i ¤˛2  gQ;˛ ; C  b˙i;Q D aQ ˚ n˙˛i ;Q ; ci;Q D n  ˛i ;Q  ˚ aQ ˚ ni;Q :  These subalgebras of gQ all satisfy .?/ and so we have the corresponding groups over Q: ˙ AQ ; BQ ; B˙i;Q ; Ci;Q ; Li;Q ; NQ ; Ni;Q ; N˙˛i ;Q ; Pi;Q :  5.2  Peter-Weyl Type Theorems 2 P, let M. /_ Q denote the subspace generated by  Definition 5.19. Given  weight vectors in the coinduced module: b  C OINDaQ Q Q where Q  D H OMUQ .a/ UQ .b/; Q  is the 1-dimensional UQ .a/-module with weight  ; . Then as UQ .b/-  modules we have: _ M. /_ Q D M.0/Q ˝Q Q :  Therefore all these modules are isomorphic as UQ .nC /-modules, in fact as UQ .nC /modules we have: C _ M. /_ Q Š UQ .n / ;  40  5.2. Peter-Weyl Type Theorems where the latter is the restricted dual with respect to the QC -grading, that is: ! M  C _  UQ .n / ´  M  C  UQ .n /˛  ˛2QC  C  UQ .n /˛  D UQ .nC / :  ˛2QC  2 Pi then  ˛ 0 and hence there exists a ; ˛i_ ˛ ˝ unique irreducible UQ .li /-module of dimension ; ˛i_ C1, which we will denote Definition 5.20. Suppose  ˝  by `i . /Q . We define Mi . /_ Q to be the subspace generated by weight vectors in the coinduced module: p  i;Q `i . /Q D H OMUQ .li / UQ .pi /; `i . /Q : C OINDli;Q  Lemma 5.21 ([17] page 25, Lemma 7). (1) For each i 2 I we have natural isomorphisms: Pi;Q D Ni;Q  Li;Q ;  BQ D Ni;Q  Bi;Q ;  Ci;Q D Ni;Q  B  C BQ D NQ  AQ :  i;Q ;  0 (2) If mQ  mQ are two subalgebras mentioned in Remark 5.18 then there  exists a natural morphism MQ ! M0Q which is a closed immersion, in other words HQ .m0 / ! HQ .m/ is a surjection of Q-algebras. (3) We have the following isomorphism: HQ .a/ D  M  Q ˝Q Q  As UQ .a/  UQ .a/-modules  2P  (5.22) HQ .li / D  M  `i . /Q ˝Q `i . /Q  As UQ .li /  UQ .li /-modules  2Pi  (5.23) HQ .nC / D M.0/_ Q  As UQ .nC /-modules  (5.24) 41  5.3. The Unipotent Subgroup HQ .b/ D  M  HQ .pi / D  M  M. /_ Q  As UQ .b/-modules  (5.25)  2P  h Mi . /_ Q  ; ˛i_ iC1  As right UQ .pi /-modules (5.26)  2Pi  Proof. (1) is easy and implies (2). For (3) we note that (5.22) and (5.23) are known from finite dimensional theory. (1) and (5.24) together imply (5.25) and (5.26). Therefore we only need to show (5.24). We claim HQ .nC / D UQ .nC /_ . First note that HQ .nC / can not be any bigger than the restricted dual since any functional not in the restricted dual is not L finite and can not be written as a finite linear combination of L -finite functionals. Pick a basis fu˛ W ˛ 2 QC g consisting of the weight vectors of the adjoint action of aQ , and let f ˛ W ˛ 2 QC g be a corresponding dual basis which spans the restricted dual. Evidently all the elements in the dual basis are L -finite, now by definition we have: ad .´/.  ˛/  .x/ D  ˛ .ad.´/.x//:  This expression is zero unless x 2 UQ .nC /˛ in which case we have: ˛ .ad.´/.x//  D  C Since nQ \ aQ D f0g we have:  ˛  ˛ .h˛;  ´i x/ D h˛; ´i  ˛ .x/:  2 HQ .nC /. Finally we note that aQ annihilates  the unit in HQ .nC /, which is the map: 1 W UQ .nC / ! Q, characterized by z Q .nC / D 0. The isomorphism between HQ .nC / and M.0/_ 1 .1/ D 1 and 1 U Q is then given by sending 1 to 10 the dual weight vector to the highest weight 10 2 M.0/Q .  5.3  The Unipotent Subgroup  C While (5.24) gives us some information about the structure of NQ in this section  we will give an explicit construction of this group. Notation 5.27. In this section UQ .nC / will be denoted by UQ .  42  5.3. The Unipotent Subgroup Definition 5.28. Consider a completion of UQ based on the root lattice decomposition: c UQ D  Y  UQ;˛  ˛2QC  Let fu1 ; u2 ;  M  UQ;˛ D UQ :  ˛2QC  g be a Q-basis for UQ with fu1 ; u2 ;  g as the corresponding dual  basis. Then from Appendices B, C and the proof of Lemma 5.21 we have: UQ Š QŒu1 ; u2 ;  ;  c UQ Š QŒŒu1 ; u2 ;  ;  _ UQ Š QŒu1 ; u2 ;  ;  UQ Š QŒŒu1 ; u2 ;  ;  _ c UQ D UQ : _ c Note that UQ ; UQ are dense subsets of UQ and UQ , respectively. C c c Definition 5.29. Let nQ be the completion of nQ in UQ and define the exponential c c map E XP W nQ ! UQ as the formal sum:  E XP.x/ D  1 X xn : nŠ  nD0  C c Lemma 5.30. NQ can be identified with E XP nQ  Proof.  1  c UQ .  C _ Since elements of NQ are Q-algebra homomorphisms, UQ ! Q, we can  C c consider NQ as a subset of UQ . _ T HE M AP : Since UQ is dense in UQ we can extend any  _ c 2 UQ to UQ by  c continuity. Now for y 2 nQ define:  ˚  _ nE XP.y/ W UQ !Q  nE XP.y/ . / D .E XP.y// _ I NJECTIVITY: This follows from UQ being dense in UQ .  H OMOMORPHISM : We may assume y to be primitive, non-primitive elements c of nQ can be expressed as limits of primitive elements. Now the following calcu1 This  proof was communicated to me by D. H. Peterson.  43  5.3. The Unipotent Subgroup lation proves the claim: .E XP.y// D E XP.y/ ˝ E XP.y/: Finally we claim that nE XP.y/ is a Q-algebra homomorphism: nE XP.y/ .  /D.  /.E XP.y//  D. ˝  /. .E XP.y//  D. ˝  /.E XP.y// ˝  .E XP.y//  D . .E XP.y// ˝ . .E XP.y// D nE XP.y/ . /nE XP.y/ . / C S URJECTIVITY: Let n 2 NQ be arbitrary, we will inductively construct y 2 c nQ such that n D nE XP.y/ , that is:  n. / D .E XP.y//; Let P 2 QŒu1 ; u2 ;  _ 8 2 UQ Š QŒu1 ; u2 ;   and assume Xk D x1 C n.P / D P .E XP.Xk //;  :  C xk is such that deg.P / Ä k:  We would like to find xkC1 that satisfies: n.P / D P .E XP.Xk C xkC1 //;  deg.P / Ä k C 1:  But based on the Campbell-Hausdorff formula we have: P .E XP.Xk C xkC1 // D P .E XP.Xk //P .E XP.xkC1 //;  deg.P / Ä k C 1;  which implies that we should have: n.P / D P .E XP.xkC1 //;  deg.P / D k C 1:  But this determines xkC1 and then we can repeat the same with XkC1 D Xk C xkC1 . Finally we take y D limk Xk . 44  5.4. The Split Maximal Kac-Moody Group  5.4  The Split Maximal Kac-Moody Group  Identify the n copies of AQ in W 1;Q ;  ; W n;Q and define W Q as the group  generated by all W i;Q subject to one additional relation. For all i ¤ j 2 I where ri rj 2 W is of finite order mij : .r i rj /mij D 1. Then we have a short exact sequence of groups: f1g ! AQ ! W Q ! W ! f1g with r i 7! ri : Definition 5.31. Identify all the copies of BQ in P1;Q ;  ; Pn;Q , then we define  the split maximal Kac-Moody group as the product of P1;Q ;  Pn;Q and W Q  amalgamated along their intersections. The resulting group is not a group scheme, it is the direct limit of groups (see Appendix D), where each group is the Q-points of a group scheme. Remark 5.32 ([17] page 27). gQ itself satisfies (?), and if dim.gQ / < 1 then one obtains Chevalley’s simply connected group. However when gQ is infinite dimensional the Hopf algebra HQ .g/ is too small to give us a suitable group. To see this let  2 HQ .g/  UQ .g/ , and consider the Q-grading: UQ .g/ D  M  UQ .g/˛ :  ˛2Q  Suppose  is non-zero on the subspace UQ .g/˛0 , however this would violate the  L -finiteness of  since there are infinitely many different ways of writing ˛0 D  ˇ0 C 0 with ˇ0 ; 0 2 Q. Therefore we denote the group defined in Definition 5.31 €Q to distinguish it from GQ discussed above. by G €F where F is any field of characteristic Remark 5.33. Similarly we may define G zero. Set: HF.m/ ´ F ˝Q HQ .m/; where mF D p1;F;  ; pn;F; aF then we have groups over F: P1;F;  ; Pn;F  and AF. The definition of W F is identical to W Q since the quotient group W does not depend on F. 45  Chapter 6  Groups over Z Starting with the group MQ defined in the previous chapter, we aim to define a group scheme over Z: MZ . In order to do so we first define a natural subring of the Hopf algebra HZ .m/  HQ .m/ (see Definition 6.1). If HZ .m/ becomes a Hopf  algebra over Z with the maps inherited from HQ .m/ (so that we have a compatible subgroup) and is a lattice in HQ .m/ (to ensure the group is large enough) we say that mQ is an integral subalgebra of gQ and define MZ to be the spectrum of HZ .m/. However not every subalgebra which satisfies .?/ (which is needed for MQ to exist in the first place) is integral so we have to impose further conditions on mQ . In §1 we introduce HZ .m/ and the concept of an integral subalagebra, in §2 we investigate which conditions are needed for mQ to be an integral subalgebra, in §3 we try to establish which subalgebras of gQ are integral and finally in §4 we define €Z in an analogous way to G €Q since the subalgebras used in the definition of G €Q G are integral as shown in §2. The material of §1 and §2 are based on [17] pages 21 - 23. It should be noted that the theory of Kac-Moody groups over integers has become a very active field of research, two recent and noteworthy references are [1, 5].  6.1  HZ .m/  Definition 6.1. Throughout this chapter mQ  gQ is a subalgebra that satisfies .?/ and hence HQ .m/ is a commutative Hopf algebra. Set: UZ .m/ ´ UZ .g/ \ UQ .m/; ˚ « HZ .m/ ´ 2 HQ .m/ W .UZ .m//  Z :  46  6.2. Integrality Conditions  Example 6.2. Let us try and compute HZ .a/. By definition HZ .a/ is the set of linear maps in HQ .a/ that take integer values when restricted to UZ .a/. Since elements of Á UZ .a/ are integral linear combination of products of elements of the _ ˛ form mi , we conclude that 2 HQ .a/ belongs to HZ .a/ if and only if .˘ _ /  Z. This combined with the map we used to show HQ .a/ D QŒP shows that HZ .a/ D ZŒP. Definition 6.3. A subalgebra mQ  gQ is called integral, if: (1) HZ .m/ is a commutative Hopf algebra with the maps inherited from HQ .m/, (2) HZ .m/ is a lattice in HQ .m/, i.e. HQ .m/ D Q ˝ HZ .m/. Definition 6.4. Given an integral subalgebra, mQ , we may define a group over Z: MZ D H OMZ  alg .HZ .m/;  Z/:  Moreover if R is any commutative ring of characteristic zero with a unit, by setting HR .m/ ´ R ˝ HZ .m/ we can define a group over R: MR D H OMR  alg .HR .m/;  R/ :  This is compatible with our earlier definition over Q and any other fields of characteristic zero.  6.2 6.2.1  Integrality Conditions Hopf Algebra  Lemma 6.5. The following hold with no extra conditions on mQ : .Z/  HZ .m/; .HZ .m//  Z; .HZ .m//  HZ .m/: 47  6.2. Integrality Conditions Proof. The unit and antipode map conditions are automatically satisfied. The conz Z .m/ ´ U z Q .m/ \ UZ .m/, which then dition for the counit map follow from U z Z .m/. implies UZ .m/ D Z ˚ U Definition 6.6. Next we present two further conditions on mQ : .UZ .m//  UZ .m/ ˝ UZ .m/;  (Ž)  8JQ 2 XQ .m/ W UZ .m/=JZ is a Z-module of finite type.  ()  And here JZ D JQ \ UZ .m/ for all JQ 2 XQ .m/. Lemma 6.7. If mQ satisfies .Ž/ then HZ .m/ is closed under multiplication. Proof. Let a; b 2 HZ .m/ and u 2 UZ .m/ then by definition of the transpose map we have: .a ˝ b/.u/ D .a ˝ b/. .u//: but .u/ 2 UZ .m/ ˝ UZ .m/ by .Ž/ and a; b 2 HZ .m/, hence: .a ˝ b/. .u// 2 Z. Lemma 6.8. If mQ satisfies ./ then HZ .m/ is closed under co-multiplication. Proof. Let  2 HZ .m/  HQ .m/, since HQ .m/ is a Hopf algebra we already  have: . / 2 HQ .m/ ˝Q HQ .m/  UQ .m/ ˝Q UQ .m/ UQ .m/ ˝Q UQ .m/ D H OMQ UQ .m/ ˝Q UQ .m/; Q  Since . / u ˝ u0 D  .u ˝ u0 / D .uu0 /;  UZ .m/, as a subring, is closed under multiplication and  (6.9)  2 HZ .m/ we have in  fact: . / 2 H OMZ UZ .m/ ˝ UZ .m/; Z :  (6.10)  48  6.2. Integrality Conditions On the other hand because  2 HQ .m/, there exists a two sided ideal JQ 2 XQ .m/  such that .JQ / D 0. Using (6.9) arrive at: . / JZ ˝ UZ .m/ C UZ .m/ ˝ JZ D 0:  (6.11)  From (6.10) and (6.11) we have:  . / 2 H OMZ  UZ .m/ ˝ UZ .m/ ;Z JZ ˝ UZ .m/ C UZ .m/ ˝ JZ  à  D H OMZ UZ .m/=JZ ˝ UZ .m/=JZ ; Z D H OMZ UZ .m/=JZ ; Z ˝ H OMZ UZ .m/=JZ ; Z By ./, UZ .m/=JZ is a Z-module of finite type, it is torsion free as well since JQ is an ideal. Therefore H OMZ UZ .m/=JZ ; Z  HZ .m/: And from this we get . /  HZ .m/ ˝ HZ .m/:  6.2.2  Lattice  Lemma 6.12. If mQ satisfies ./ then HQ .m/ D Q ˝ HZ .m/. Proof. If  2 HQ .m/ then there exists a two sided ideal JQ 2 XQ .m/ such that  .JQ / D 0. Therefore we have a homomorphism of additive abelian groups: W UZ .m/=JZ ! Q: However UZ .m/=JZ is a Z-module of finite type and so the image of  is fully  determined by its value on the finitely many generators of UZ .m/=JZ . By taking the least common denominator of the image of this finite set of generators we see that there exists 0 ¤ p 2 Z such that: .UZ .m//   1 p Z:  49  6.3. Integral Subalgebras We therefore have p 2 HZ .m/.  6.3  Integral Subalgebras  Theorem 6.13. If a subalgebra mQ  gQ satisfies .?/; .Ž/ and ./ then it is integral. Proof. This follows from Lemmas 6.7, 6.5, 6.8 and 6.12. Using Theorem 6.13 in this section we examine whether the well known subalgebras of gQ are integral or not. However before doing so we find an equivalent, but easier to verify, statement for ./. We start with a definition: Definition 6.14. Let YQ .m/ be the set of all UQ .m C a/-modules, E, such that: (1) E is finite dimensional. (1) E is a weight module for aQ . (2) weights of E lie in P. Lemma 6.15. ./ is equivalent to: 8E 2 YQ .m/; 8e 2 E W  UZ .m/ e is a Z-module of finite type.  (ŽŽ)  Proof. If JQ 2 XQ .m/ then UQ .m/=JQ 2 YQ .m/. .ŽŽ/ implies that UZ .m/ .x C JQ / is a Z-module of finite type for all x 2 UQ .m/. In particular UZ .m/ .1 C JQ / is a Z-module of finite type. But since UZ .m/ acts on UQ .m/=JQ by left multiplication and so UZ .m/ .1 C JQ / Š UZ .m/=JZ which proves ./. Assume ./ and let E 2 YQ .m/ be arbitrary. Now the set of elements e 2 E such that UZ .m/ e is a Z-module of finite type, is itself a UQ .m C a/-submodule of E. Lemma 6.16. (1) If aQ C mQ satisfies .ŽŽ/ then mQ also satisfies .ŽŽ/. C (2) If tQ D aQ ˚ mQ then UZ .t/ D UZ .mC / ˝ UZ .a/.  50  6.3. Integral Subalgebras C C (3) If tQ D aQ ˚ mQ then tQ and mQ both satisfy .ŽŽ/.  (4) If aQ  mQ and UZ .m/ D UZ .m / ˝ UZ .t/, then mQ satisfies .ŽŽ/. Proof. Since UZ .m/  UZ .a C m/ Lemma 6.15 implies (1) at once. For each ˇ 2 Q: UZ .mC /ˇ D UQ .mC /ˇ \ UZ .nC /ˇ . Since UZ .nC /ˇ is a Z-module of finite type, UZ .mC /ˇ is a direct factor of UZ .nC /ˇ . Hence UZ .mC / is a direct factor of UZ .nC /. From this and UZ .b/ D UZ .nC / ˝ UZ .a/ we get (2). Using part (1) we see that to show (3) it suffices to show that tQ satisfies .ŽŽ/. So let E 2 YQ .t/ and e 2 E, we can also assume that e is a weight vector and therefore: UZ .a/ e D Ze. Now using part (2) we may write: UZ .t/ e D UZ .mC / ˝ UZ .a/  e D UZ .mC / .Ze/:  Since E is finite dimensional there exists a finite set ˙  QC such that: UZ .mC / e D  M  UZ .mC /˛ e:  ˛2˙  Since UZ .mC /˛ are Z-modules of finite type this proves 3. For (4) let E 2 YQ .m/ by (3), for every e 2 E the Z-modules UZ .m / e and UZ .t/ e are of finite type. Now UZ .m/ D UZ .m / ˝ UZ .t/ implies (4). Remark 6.17. The Lie algebras enumerated in Remark 5.18 are integral, we already know that all these subalgebras satisfy .?/. For .Ž/ we note that it is enough to show it for the generators of each subalgebra. Finally for ./ we use the equivalent formulation .ŽŽ/ and Lemma 6.16. So we have the corresponding groups over Z: ˙ AZ ; BZ ; B˙i;Z ; Ci;Z ; Li;Z ; NZ ; Ni;Z ; N˙˛i ;Z ; Pi;Z :  Lemma 6.18.  51  6.4. The Arithmetic Group (1) For each i 2 I we have the following isomorphisms: Pi;Z D Ni;Z  Li;Z ;  BZ D Ni;Z  Bi;Z ;  Ci;Z D Ni;Z  B  C BZ D NZ  AZ :  i;Z ;  0 (2) If mQ  mQ are two subalgebras enumerated in Remark 5.18 then there  exists a natural morphism MZ ! M0Z which is a closed immersion. (3) For each i 2 I the natural morphisms N  ˛i ;Z  N˛i ;Z  BZ ! Pi;Z Ci;Z ! Pi;Z  are open immersions. Proof. The only delicate point is to show (2). More precisely, to show that BZ ! Pi;Z is a closed immersion. Using isomorphism of groups Pi;Z D Ni;Z Li;Z and BZ D Ni;Z  Bi;Z , we reduce to show that Bi;Z ! Li;Z is a closed immersion,  this is done by direct calculation. We show (3) with the same argument.  6.4  The Arithmetic Group  Definition 6.19. Now that we have the required group schemes, we define, € , the split maximal Kac-Moody group over Z as the product of P1;Z ;  ; Pn;Z and W Z ,  amalgamated along their intersections. As before (see Definition 5.31) this group itself is not a group scheme over Z.  52  Chapter 7  Structure Theory €F acts on any integrable irreducible highest weight module In §1 we show that G L. /F. In fact we give an explicit construction for the representation map which is based on the last paragraph of page 28 in [17]. Next we see how different subgroups act and what their matrices look like in an admissible basis, most importantly we show that the Chevalley lattice is stable under € (see Lemma 7.9). €F possesses a Tits system, this In §2 we use the representations to show that G €F and follows §5.12 in [23]. proof is based on the representation theory of G In §3 we prove an Iwasawa decomposition when F D R; C. This prove uses the existence of the Tits system shown earlier, with Lemma 7.26 on the theory of Tits systems being the crucial part of the proof. In §4 we introduce the minimal group of Kac and Peterson, we will use this group to show that the orbit of the highest weight vector is not the entire module.  7.1 7.1.1  Representation Theory Constructing the Map  Let L. /F be an integrable highest weight module for gF. We wish to construct a representation: €F ! GL.L. /F/: G €F it suffices to do so for each minimal parabolic subgroup By the construction of G Pi;F. For every i 2 I , since UF.pi / acts on L. /F, every v 2 L. /F gives rise to a map  v  W UF.pi / ! L. /F defined by  integrable we see that all  v  v .x/  D x v. Since L. /F is  have finite rank. But on the other hand we have a  53  7.1. Representation Theory F-vector space isomorphism: H OMfin F .UF .pi /; L. /F / Š UF .pi / ˝F L. /F ;  (7.1)  where H OMfin F . ; / is the set of all F-linear maps of finite rank. So combining v 7!  v  with this isomorphism gives us a map: L. /F ! UF.pi / ˝F L. /F:  (7.2)  We claim that the image of the map in (7.2) in fact lies in HF.pi / ˝F L. /F. To see this we first recall the isomorphism of (7.1) can be explicitly written as: 1 X « W UF.pi / ! L. /F ! 7 xi ˝  ˚  .xi /  i D1  where fx1 ; x2 ;  g is a basis for UF.pi / of our choosing. We take it to be the basis  afforded to us by the PBW Theorem with the ordering: e  i  < ˛1_ <  < ˛n_ < e1 <  < en <  If we show that the elements of the dual basis fx1 ; x2 ;  :  g all lie in HF.pi /, we  are done. But then we may write: HF.pi / D HF.ni / ˝F HF.li / and we know that HF.ni / and HF.li / are the restricted dual of the corresponding universal enveloping algebras. Hence the same has to be true for HF.pi /. So we have a map: L. /F ! HF.pi / ˝F L. /F: From the theory of affine group schemes this corresponds to a representation of the group: Pi;F ! GL.L. /F/: €F Remark 7.3. The above construction more than giving us a representation for G has the virtue that it enables us to actually compute the representation for the 54  7.1. Representation Theory parabolic subgroups. Suppose we are given p 2 Pi;F then we may write its action based on the following sequence using the fact that p 2 Pi;F is in fact a homomorphism of F-algebras: p W HF.pi / ! F one can write: L. /F  / HF .pi / ˝F L. /F p˝1 / F ˝F L. /F  Š  / L. /F ;  So for a given v 2 L. /F we may write: p v D .p ˝ 1/  1 X  ! mi ˝ .mi v/  i D1  D D  1 X i D1 1 X  p mi ˝ .mi v/ p mi .mi v/  i D1  where fm1 ; m2 ;  g is a basis of our choosing for UF.pi / and fm1 ; m2 ;  g is its  dual.  7.1.2  Subgroups  Remark 7.4. Since 1 is the unit of HF.pi / and p is a F-algebra homomorphism we have: p.1 / D 1 for all p 2 Pi;F. So if we choose a basis such that m1 D 1 we can write the action in a slightly more useful form: p vDvC  1 X  p mi .mi v/:  i D2  Lemma 7.5. In any admissible basis for L. /F the elements of AF are represented by diagonal matrices. More precisely a 2 AF acts on L. /F; by a . Proof. This follows from AF being the same as classically defined torus. C Lemma 7.6. In any admissible basis for L. /F the elements of NF are repre-  sented by upper triangular unipotent matrices. In particular: n 1 n2  D 1 for all  C NF .  55  7.1. Representation Theory Proof. Let v 2 L. /F be a weight vector arbitrary, we have: n vDvC  1 X  n mi .mi v/  i D2  But when acting on L. /F any non-constant monomial in UF.nC / will lower depth. So the terms in the sum (which are finitely many) are vectors in L. /F and all of their weight components have lower depth than v. C C C . AF D AFNF , in particular BF D NF Corollary 7.7. AF normalizes NF  Proof. This follows from Lemmas 7.5 and 7.6. C Lemma 7.8. NF acts faithfully on non-trivial integrable highest weight modules. C Proof. Suppose that there exists n 2 NF such that n v D v for all v 2 L. /F.  Based on the action of n this means: 8v 2 L. /F W  1 X  n mi .mi v/ D 0:  i D2  Now we can find a vector v2 2 L. /F such that m2 v2 ¤ 0 but mi v2 D 0 for all i > 2. This shows that n m2 D 0. By repeating this argument we see that n mi D 0 for all i > 2.  7.1.3  The Arithmetic Subgroup and the Chevalley Lattice  Lemma 7.9. For all p 2 Pj;Z we have: p L. /Z  L. /Z .  Proof. Let v 2 L. /Z be arbitrary, then we have: p vD  1 X  p.mi /.mi v/:  i D1  The series is in fact a finite sum we just need to show that each term belongs to L. /Z . Since mi 2 UZ .pj / we have: mi v 2 L. /Z , that leaves p.mi /. However p 2 Pj;Z is a ring homomorphism: p W HZ .pj / ! Z and since HZ .pj / is a lattice in HF.pj / we can always choose a basis such that mi 2 HZ .pj /. 56  7.2. Existence of a Tits System  7.2  Existence of a Tits System  Notation 7.10. For w 2 W , we use w to denote an element in W F which lies in the coset wAF and under the quotient map W F ! W goes to w. Definition 7.11. For any real root  D w.˛j / we define N  ;F  ´ wN˛j ;Fw  1  .  Lemma 7.12. Let Z D P1;F [ [ Pn;F [ W F by definition there is a canonical €F. This map is injective. map Z ! G Proof. When acting on L. /F the image of Z under the canonical map acts as Z itself would. So if we find a module VF on which the elements of Z act faithfully, we are done. Let f$1 ;  ; $n g be a basis consisting of dominant integral weights  for P and set: VF ´ L.$1 /F ˚ AF acts faithfully since f$1 ; C NF  ˚ L.$n /F:  ; $n g generate P. By Lemma 7.8 we know that  acts faithfully on each integrable highest weight module and therefore it acts  faithfully on VF. Given an admissible basis for each module, L.$i /F, AF acts C through diagonal matrices and NF acts through upper triangular unipotent matri-  ces. Hence BF also acts faithfully. The faithfulness of the action of Pi;F follows from this and the simplicity of P GL2;F. This leaves W F, suppose there exists w 2 W F that acts trivially on VF. Since AF acts faithfully we can also assume that w ¤ 1. Therefore there exists such that w. / 2 N  ;F  but wN  of Nw.  /;F  ;F w  . Since w acts trivially the action of wN w 1  D Nw.  /;F .  However w. / 2  1  2  re C  is the same as  which means the elements  are represented by lower triangular unipotent matrices in GL.VF/.  This contradiction completes the proof. Remark 7.13. Let NF;min denote the subgroup that is generated by the collection of ˚ « the subgroups: N ;F W 2 re . In an admissible basis for VF elements of BF (resp. NF;min ) operate through upper triangular matrices (resp. unipotent lower triangular matrices). In particular we have: BF \ NF;min D f1g:  57  7.2. Existence of a Tits System We use the superscript to emphasize the fact that NF;min ¤ NF . In fact NF is not €F . even a subgroup of G C Remark 7.14. We identify the groups AF; NF ; BF; W F; Pi;F with their images €F . in G  €F . Lemma 7.15. BF and W F generate G Proof. This follows from the fact that each Pi;F is generated by BF and W i;F. Lemma 7.16. BF \ W F D AF. Proof. We know AF 2  re C  BF \ W F. Let w 2 BF \ W F be such that w ¤ 1. Pick  such that w. / 2 wN  Since w 2 BF and N  ;F  wN  . Then: ;F w  1  D Nw.  /;F  NF;min :  BF we have: ;F w  1  BF \ NF;min D f1g;  C €F . which contradicts the faithfulness of NF !G  Lemma 7.17. r i BFr i ¤ BF. Proof. This follows from: r i N˛i ;Fr i D N  ˛i ;F  NF;min and NF;min \ BF D  f1g. Lemma 7.18. Pi;F D BF [ BFr i Bi;F. Proof. Recall that Li;F is a finite dimensional reductive group of semi-simple rank 1 and therefore it has a Bruhat decomposition: Li;F D Bi;F [ Bi;Fr i Bi;F. Combining this decomposition with the isomorphisms: Pi;F D Ni;F BF D Ni;F  Li;F and  Bi;F, which we have from Lemma 5.21 completes the proof.  Lemma 7.19. For all w 2 W we have: r i BFw  BFwBF [ BFr i wBF.  Proof. We divide the proof in two cases:  58  7.3. Iwasawa Decomposition P OSITIVE C ASE : suppose C NF  1 .˛ / i  Dw  0 then w  1  N˛i ;Fw D N  ;F  and we have: C r i BFw D r i AFNF w C D r i AFr i r i NF w  D AFr i Ni;FN˛i ;Fw D AFr i Ni;Fr i r i N˛i ;Fw D AF r i Ni;Fr i r i N˛i ;Fw D AFNi;Fr i N˛i ;Fw D AFNi;Fr i ww D AFNi;Fr i wN  1  r i Ni;Fr i  Ni;F  N˛i ;Fw  ;F  BFr i wBF: N EGATIVE C ASE : suppose  Dw  1 .˛ / i  0, then .ri w/  1  .˛i /  0 and  we have: r i BFw D .r i BFr i / r i w Pi;Fr i w .BF [ BFr i BF/ r i w  Lemma 7:18  BFr i w [ BF .r i BFr i / w BFr i w [ BFBFr i .r i w/BF  Positive Case  BFr i w [ BFwBF BFr i wBF [ BFwBF: Theorem 7.20.  €F; BF; W F; fr 1 ; G  Á ; r n g is a Tits system.  Proof. This follows from Lemmas 7.15, 7.16, 7.17 and Corollary 7.19.  7.3  Iwasawa Decomposition  Notation 7.21. In this section F D R; C. 59  7.3. Iwasawa Decomposition Definition 7.22. Let  Ž  denote the Hermitian conjugate with respect to positive  definite inner product f ; g defined on L. /F in §3.4. Based on the construction Ž of f ; g we have: ei D e i for all i 2 I . A linear operator T W L. /F ! L. /F is called unitary if T Ž D T  1.  A group of linear automorphisms of L. /F is  called unitary if all its elements are unitary operators. €F be the subgroup consisting of all the elements g 2 Definition 7.23. Let K G €F such that g Ž is defined and equals g 1 , we refer to this subgroup as the unitary G €F . form of G Remark 7.24. Note that K is non-trivial, it clearly contains Ki , the real maximal compact subgroup of Li;F since L. /F decomposes as a direct sum of irreducible representations of li;F. Moreover since r i 2 Ki we see that W  K.  Lemma 7.25. BFr i BF D BFr i N˛i ;F. Proof. First we note the following: r i BFr i D r i Ni;FN˛i ;FAF r i D Ni;FN  ˛i ;F AF  D Ni;FAFN  ˛i ;F :  Then we can write: BFr i BF D BFr i BF .r i r i / D BF Ni;FAFN D BF N  ˛i ;F  ri  ˛i ;F r i  D BFr i N˛i ;F Next we need a Lemma from the theory of Tits systems: Lemma 7.26 (Proposition 3.1 [14]). Let .G; B; N; S / be a Tits system with Weyl group W D N=.B \ N /. If w1 ; w2 2 W satisfy `.w1 w2 / D `.w1 / C `.w2 /, and if X1 ; X2 are subsets of G satisfying: (1) Bw1 B D X1 B with uniqueness of expression, (2) Bw1 B D X2 B with uniqueness of expression. 60  7.3. Iwasawa Decomposition Then: Bw1 w2 B D X1 X2 B with uniqueness of expression. €F D BFK. Proposition 7.27. G Proof. By Bruhat decomposition we only need to prove the claim for a Bruhat cell: BFwBF. By Lemma 7.26 it is enough to do so for the Bruhat cells corresponding to the fundamental reflections: BFr 1 BF; we only need to show that r i N˛i ;F r i N˛i ;F  ; BFr n BF. But using Lemma 7.25  BFK. But in Li;F we already have that  Bi;FKi D Li;F since Bi;F  BF this completes the proof.  1 Definition 7.28. Let AC F (resp. AF ) be the subgroup of AF consisting of elements  whose eigenvalues in L. /F are positive real numbers (resp. are modulus one). 1 Then we have a polar decomposition: AF D AC F AF . C C Lemma 7.29. NF AF \ K D f1g. C C Proof. When acting on L. /F the elements of NF AF \ K are represented by  unitary upper triangular matrices in any admissible basis. Therefore they have to C C be diagonal so NF AF \ K  C AC F but the only unitary element of AF is the  identity. €F D N C AC K with uniqueness of expression. Iwasawa Decomposition. G F F €F D BFK and observing that: Proof. The decomposition follows from G C C 1 C C BF D NF AF AF D NF AF .K \ BF/:  €F has two decompositions: To prove the uniqueness suppose g 2 G g D nak D n0 a0 k 0 : Then: k0k  1  Da  1  n  1 0 0  Da  1  n  1 0  D a  1  n  na  n aa  1 0  1  na a  a0 1 0  a  D n00 a00 61  7.4. The Orbit of the Highest Weight Vector Therefore k  1k0  C C AF \ K D f1g. This implies: k D k 0 . But since we also 2 NF  C 0 0 \ AC have NF F D f1g we can deduce that a D a and n D n .  €F we use gN gA gK to denote its Iwasawa decomposition. Notation 7.30. For g 2 G Remark 7.31. Let K0 denote the group generated by K1 ;  ; Kn , this is a subgroup  of K. However the proofs given in the section work with K0 instead of K as well €F D N C AC K0 with uniqueness of expression, which proves and one obtains G F  F  K D K0 .  Remark 7.32. The article [8] provides a different proof of Iwasawa decomposition for split Kac-Moody groups.  7.4  The Orbit of the Highest Weight Vector  €F. But In this section we will look the orbit of 1 2 L. /F under the action of G before doing so we need to introduce the minimal group: For each i 2 I there exists a subgroup Gi;F gi;F D n  ˛i ;F  Li;F with Lie algebra:  ˚ F˛i_ ˚ n˛i ;F  li;F:  In fact we have an isomorphism «i W SL2;F ! Gi;F. Notation 7.33. For convenience we will use the following notation: a b c d  ! ´ «j j  a b  How the groups G1;F  2 Gj;F:  c d  Then one sees that: rj D  !!  1 1  ! : j  ; Gn;F interact follows from the relations defining the  Kac-Moody algebra gF (see §2 in [14] for details): Lemma 7.34. (1) The torus, AF, is the product of the tori in G1;F  ; Gn;F. 62  7.4. The Orbit of the Highest Weight Vector (2) For i; j 2 I and t 2 F : !  t t  a b  1  !  c d  i  1  !  t  D  1  t  j  t Aij b  a Aij  t  i  c  d  ! : j  (3) For i; j 2 I; i ¤ j and u; v 2 F: 1 u  !  1  !  1 v 1  i  !  1  D  v 1  j  1 u  !  1  j  : i  (4) For i; j 2 I and t 2 F : rj  !  t t  rj 1 D  1  !  t  i  t  t  !  Aij  1  t Aij  i  : j  ; Gn;F was first introduced in [14]. We will  The group generated by G1;F  KP refer to it as the minimal group and denote it by GF . In [19] it is shown that the  orbit of the highest weight vector in the projective space, P .L. /F/, is given by quadratic equations. More precisely: KP Theorem 7.35. All v 2 GF 1 satisfy the following in L. /F ˝F L. /F:  . j /v ˝ v D  X  x˛.i / v ˝ y˛.i / v;  ˛2 [f0g  n o n o .i / .i / where x˛ and y˛ are dual basis for g˛;F and g  ˛;F  with respect to the  standard invariant form on gF. €F, moreover when F D R; C we have The minimal group is a subgroup of G €F is the unipoAC ; K G KP therefore the only difference between G KP and G F  F  F  tent group, where the minimal group is missing all the positive imaginary roots. However due to the structure of the highest weight modules the action of the two unipotent groups and hence the two groups are the same.  63  Chapter 8  Reduction Theory Notation 8.1. In this chapter we work only over the real numbers so we will drop all the subscripts. Let G be a finite dimensional real reductive group, V a representation for G and a lattice VZ length in g  1  V , reduction theory is concerned with vectors of minimal  VZ as g varies in G. However using such an approach in the  symmetrizable indefinite case immediately runs into difficulties. First since V is infinite dimensional there might not be a positive lower bound for the vectors in g  1  VZ , in other words one might have an infinite sequence of vectors in g  1  VZ  whose length approaches zero (an explicit example is given in Remark 8.3 below). So then the question becomes what is the right subset to consider, in other words before proving anything we have to find the right question first. In §1 we introduce four different subsets that have the necessary conditions to possess a reduction theory. € AR which is the most natural choice from a geometric point In §2 we define G € However at present we don’t have a reof view and contains a large subset of G. € AR since it is not clear whether points on G € AR do have minima duction theory for G on their € -orbits. € MO , the elements of which all have minima on In §3 we define another subset, G their € -orbits. Then Borel’s proof of reduction theory from the finite dimensional case ([3] 16.6) works without any changes. € MO contains a large subset of the group: N C E XP.I NT.T //K In §4 we show G (Theorem 8.23). The proof of this theorem is done in two parts: in §4.1 we give a spectral characterization of N C E XP.I NT.T //K. Then in §4.2, using this spectral € MO . The characterization, we show that N C E XP.I NT.T //K is indeed a subset of G € [ ) was inspired by the main idea of defining elements with decay (denoted by G proof of Lemma 17.15 in [12]. 64  8.1. The Four Subsets In §5 using direct calculations in the Kac-Moody group corresponding to a rank € [ is not €-invariant. 2 GCM we show that G Finally in §6 we list a number of open problems.  8.1  The Four Subsets  ˛ ˝ be the weight defined by ; ˛i_ D 1 for all i 2 I . We fix € where it acts from the right. L. / as a representation of G Notation 8.2. Let  Since L. / is infinite dimensional having a positive lower bound does not ensure the existence of a minimum, moreover since € 1 ¨ L. /Z (see §7.4) one € as candidates to work with: can consider four different €-invariant subsets of G (1) inf (2)  g  2€  g  1  1  1  1  1  achieves a minimum as  (3) infv2L. (4) Let S1 ;  g  1  /Z  g  1  1  > 0. varies in € .  v > 0.  v achieves a minimum as v varies in L. /Z .  € The relationship between these ; S4 be the corresponding subsets in G.  sets are summarized in the following diagram where arrows indicate inclusion:2 S4  / S3  > S1  S2 2 in  the following simplified form it becomes easier to order the sets: let f W X ! R>0 be a function and let Y ¨ X. Now compare the following statements: (1) f has a positive infimum on Y , (2) f has a minimum on Y , (3) f has a positive infimum on X and (4) f has a minimum on X .  65  8.2. The Arithmetic Set  8.2  The Arithmetic Set  In this section we consider the set S3 . One can formulate the definition of S3 in an € defines a new metric on L. /: equivalent but more intuitive way: any element of G 1  kvkg ´ g  v :  € is an arithmetic point if there is positive lower bound for the length We say g 2 G of the elements of L. /Z under the metric k kg . The set of all arithmetic points € AR to denote the arithmetic is called the arithmetic set. From now we will use G subset instead of S3 . Remark 8.3. This definition is based on the finite dimensional theory, in fact in € AR is the entire group (see [3] 16.2). This is not the case here, since that case G the torus, AC , has non-arithmetic points: let a 2 AC be such that a˛i < 1 for all ˚ «1 i 2 I . Consider the sequence 1 k kD1 , where dp. k / ! 1 as k ! 1, then: 1  a  1  k  Da  k  Da  a  k  p1k  a˛1  Da  a˛n  pnk  Á  :  Since ht. k / ! 1 as k ! 1 and a˛i < 1 for all i 2 I we see that: lim  k!1  a  1  1  k  D 0:  € and v 2 L. / with the weight space decomposition: Lemma 8.4. Let g 2 G v D c1 1 Then: g  1  v  gA  j0  1  C  ˇ ˇ ˇcj ˇ, where 0  C ck 1 j0  2f  k  ; 1;  cj 2 R: ;  kg  is of maximal depth in  that set. Proof. g  1  v D gA1 gN1 v gA1 D  gA  K is unitary Á  cj0 1 j0 Á j0 cj0 1 j0  choice of  j0  66  8.2. The Arithmetic Set D gA  j0  gA  j0  ˇ ˇ ˇcj ˇ 1 0 ˇ ˇ ˇcj ˇ 0  gA  j0  1  j0  j0  >0 D1  The following Lemma shows why the set of arithmetic points of the Torus, which we will denote by AAR , is important: Lemma 8.5. N C AAR K  € AR . G  € be such that gA 2 AAR we aim to show that g 2 G € AR . Since the Proof. Let g 2 G ˚ « weight vectors 1 W 2 P are a basis for L. /Z in order to show that there is a positive lower bound when g acts on L. /Z we only need to show g has a lower bound as it acts on these vectors. We have: g  1  D gA1 gN1 1  1  D gA1  gN1 1  Now let v D gN1 1 , since elements of N C are represented by unipotent upper triangular matrices in GL.L. // the weight space decomposition of v can be written as follows: v D 1 C c1 1 where  1  C  C ck 1  has maximal depth among the weights f ;  k  ;  1;  k g.  ;  Using Lemma 8.4  with gA and v yields: gA1 v  gA :  In summary we have shown: g  1  1  gA1 1  D gA ;  However since gA 2 AAR the right hand side has a positive lower bound. Remark 8.6. A converse to Lemma 8.5 does not hold, in other words it is possible € AR and gA … G € AR . First we need to do a calculation: to have g 2 G g  1  1  2  D gA1 gN1 1 D gA1 1  2  2  C gA1 v  2  67  8.2. The Arithmetic Set 2  X cj 1 C gA1 X 2 C cj2 gA j  D gA  2  D gA  Á  2  j  gA … AAR therefore: gA ! 0 as dp. / ! 1. However there are at least two ways of avoiding a contradiction: 1  there exists M 2 N such that n  0  < M for any weight vector 1 2 L. / .  0/  with dp.  1 has a weight component in L. /  the matrix coefficients of n  1  2 GL.L. // grow exponentially with depth.  Because of Lemma 8.5 next we will try to compute AAR . Definition 8.7. We define the following subset of positive roots: re  im  and  D  ˚  re C  2  « 2P :  W  are defined similarly. D  Lemma 8.8.  C. re  Proof. We show this by proving ing coroot as  D  re C  im  and  w.˛i_ /.  im C.  Now using Proposition 11.1 in [15] we have:  re Cn  re  D  ˚  2  re C  W  ˝  ;  However we see that the set is in fact empty since since  D  D w.˛i / be a positive real root, we define its correspond-  R EAL ROOTS : Let _  D  is a positive root,  _  _  ˛  « D0 ; is dominant and integral and  will be a positive coroot.  I MAGINARY ROOTS : Using Corollary 11.9 [15] im D P . j / ¤ 0 for all D pi ˛i 2 C , however (§5.2 [15]): . j /D  n X i D1  pi . j˛i / D  n X  pi  2 1 2 j˛i j  ˝  i D1  ˛ ; ˛i_ D  1 2  n X  im C  is equivalent to  pi j˛i j2 > 0:  i D1  Lemma 8.9. E XP.T /  AAR  E XP.C L.T //. 68  8.3. Points with Minima Proof. Suppose E XP.´/ D a 2 AAR , using Lemma 8.8 we have: 0 < inf a 2P  a D inf a D inf e h  D inf a 2  2  2  ; ´i  D inf e h 2  ; ´i  C  Which is equivalent to: 9L 2 R W 8 2 Now recall that if  im C  2  then n  C  W h ; ´i  im C  2  L:  for all n 2 N. Therefore we can  conclude: 8 2  im C  W h ; ´i  0:  by Proposition 2.48 this means ´ 2 C L.T /. Conversely let a D E XP.´/ 2 E XP.T /. Since the weight vectors generate the Chevalley lattice so it is enough to show: 0 < inf a 2P  :  However based on Bardy there exist w0 2 W ; ´0 2 D such that ´ D w0 .´0 /, therefore we have: a  D eh  where  2  as long as  ; ´i  D eh  C . Now w0 1 . / 2  ; w0 .´0 /i  D e hw0  1  .  /; ´0 i  1 D e hw0 .  since ´0 2 D we have h˛i ; ´0 i C  there is a lower bound for a  /; ´0 i  e hw0  1  . /; ´0 i  ;  0 for all i 2 I . Therefore . Now the result follows  from ˇ ˇw  0  1  .  C/  \  ˇ ˇ < 1:  € AR . Corollary 8.10. N C E XP.T /K  G Proof. This follows from Lemma 8.5 and Lemma 8.9.  8.3  Points with Minima  The problem with the arithmetic set is that in order to prove the reduction theorem one needs to first show the existence of minima. That is, for a given point g the 69  8.3. Points with Minima function ˚ achieves a minimum on the set € g. However since L. / is infinite dimensional it is not obvious that an arithmetic point would achieve its minimum, we might have a situation where the infimum is not in fact a minimum. Therefore we consider the subset S2 instead. € ! R>0 given by: ˚.g/ D Definition 8.11. First we define a function ˚ W G € such that ˚ achieves a g 1 1 . Then S2 is the set of all elements g 2 G, positive minimum when considered as a function on the orbit: €g. More formally: n €W 9 S2 D g 2 G  0  o 2 € ; 8 2 € W ˚. 0 g/ Ä ˚. g/ :  € MO to denote S2 . From now on we will use G € MO is a €-invariant subset of G. € In fact if one defines: Remark 8.12. By design, G n o € W 8 2 € W ˚.g/ Ä ˚. g/ ; D g2G € MO D € . then: G € MO is in fact non-empty, Remark 8.13. It is not obvious from the definition that G € (see 8.23). below we will show that it does contain a large subset of G Definition 8.14. For  > 0 define: ˚ A D a 2 AC W 8i 2 I W a˛i  «  :  € to denote the set N C A K. We also use G Reduction Theorem. There exists a real constant € MO there exists 2 € with g 2 G € . G  8.3.1  > 0 such that for any g 2  Proof of Reduction Theorem  € MO D € , it is enough to show that Remark 8.15. Since G constant  € for some real G  > 0.  70  8.3. Points with Minima € D Ni Ai Li K, where Ai D K ER.˛i / Remark 8.16. For each i 2 I we can write G € has a decomposition: A. In particular any g 2 G g D ng ag lg kg ; where ng 2 Ni ; ag 2 Ai ; lg 2 Li and kg 2 K. Lemma 8.17. ˚ is left-invariant under N C and right-invariant under K, in fact we have: ˚.g/ D gA . Proof. Using Iwasawa decomposition, N C 1 D 1 and the fact that K is unitary with respect to k k we see that ˚.g/ D ˚.N C g/ D ˚.gK/ and the first assertion is proven. Now we can write: ˚.g/ D ˚.gA / D gA1 1  D gA 1  Now recall that k k is normalized such that 1  D gA  1  :  D 1.  Lemma 8.18. ˚.g/ D ˚.ag /˚.lg /. Proof. Using the decomposition from Remark 8.16 and then applying Lemma 8.17 we see that ˚.g/ D ˚.ag lg /, now we use the definition of ˚: ˚.ag lg / D lg 1 ag 1 1 Lemma 8.19. If Proof.  D ag  lg 1 1  D ˚.ag /˚.lg /:  2 € \ Li then: ˚. g/ D ˚.ag /˚. lg /.  2 Li therefore it normalizes Ni and commutes with Ai : ˚. g/ D ˚  ng ag lg kg D ˚ n0g ag lg kg D ˚.ag lg /;  now we use the previous Lemma. Lemma 8.20. If g 2  then: ˚.lg / Ä inf  2€\Li  ˚. lg /.  71  € MO 8.4. A Subset of G Proof. ˚.ag /˚.lg / D ˚.g/  Lemma 8.18  Ä inf ˚. g/  g2  Ä  2€  D  inf  ˚. g/  € \ Li  inf  ˚.ag /˚. lg /  Lemma 8.19  2€\Li 2€ \Li  D ˚.ag /  inf  2€ \Li  €  ˚. lg /  Cancelling ˚.ag / from both sides gives us the result. Lemma 8.21. There is a real constant, " > 0, such that: inf  2€\Li  ˚. lg / Ä ".  Proof. This follows from Li being a finite dimensional reductive group of semisimple rank 1. Lemma 8.22. If g 2  then for all i 2 I we have: ˛i lg;A  "  2  :  Proof. Based on Lemma 8.21 and Lemma 8.20 we have: "  inf  2€\Li  ˚. lg /  ˚.lg / D lg;A :  Now note that jLi D 12 ˛i . Finally Lemma 8.22 combined with the following observation completes the proof: ˛i gA D lg;A ag  8.4  ˛i  ˛i D lg;A :  € MO A Subset of G  In this section we prove the following: Theorem 8.23. N C E XP.I NT.T //K  € MO . G 72  € MO 8.4. A Subset of G  8.4.1  A Spectral Characterization of I NT.T /  Definition 8.24. For a 2 AC and C > 0 define: P .a; C / ´  ˚  2P W a  « C :  Definition 8.25. We say a 2 AC decays in L. / if P .a; C / is finite for all C > 0. Notice that if a decays in L. / then a ! 0 as dp. / ! 1 for The subset of  AC  Lemma 8.26. A[  consisting of all such elements is denoted by  2P.  A[ .  E XP.I NT.T //  Proof. Let E XP.´/ D a 2 A[ , then one has the following: ˇn ˇ ˇ ˇ 8C > 0 W ˇP .a; C /ˇ < 1 $ 8C > 0 W ˇ ˇn ˇ $ 8C > 0 W ˇ ˇn ˇ $ 8C > 0 W ˇ ˇn ˇ $ 8C 0 > 0 W ˇ ˇn ˇ $ 8C 0 > 0 W ˇ ˇ˚ $ 8K 2 R W ˇ  oˇ ˇ C ˇ<1 oˇ ˇ W a C ˇ<1 oˇ ˇ W a Äa C 1 ˇ<1 oˇ ˇ W a Ä C0 ˇ < 1 oˇ ˇ W e h ; ´i Ä C 0 ˇ < 1 «ˇ W h ; ´i Ä K ˇ < 1  2P W a 2 2 2 2 2  Taking K D 0 and using Lemma 8.8 implies ´ 2 I NT.T /. Lemma 8.27. A[ is invariant under the action of W . Proof. Let E XP.´/ D a 2 A[ and take C > 0 and w 2 W to be arbitrary: P waw  1  ;C D  n  2 P W waw  D  n  2 P W E XP .w.´//  D  n  2 P W eh  D  n  2 P W e hw  D  n  2 P W aw  1  C  ; w.´/i  1.  C C  /  C  o  o  /; ´i  1.  o  C  o  o  73  € MO 8.4. A Subset of G And this last set is finite because a 2 A[ and W permutes the set of weights. A[ .  Lemma 8.28. E XP .Dfin /  Proof. Let ´ 2 Dfin then by definition ´ 2 FJ , where J  I is an arbitrary subset  of finite type. Take M´ to be the minimum of the set fh˛i ; ´i W i … J g, note that re C  2  M´ is a positive real number. For any  one sees that h ; ´i  htJ . /M´ ,  where htJ is defined as follows: htJ Therefore for  X i 2I  Á X pi ˛i D  i …J  pi :  2 P one has:  h ; ´i D h  ; ´i D h ; ´i  h ; ´i Ä h ; ´i  htJ . /M´ :  And so for a D E XP.´/ we can write: a D eh  ; ´i  Ä eh  ; ´i  e  htJ . /M´  :  Let C > 0 be arbitrary, then: C Ä a Ä eh  ; ´i  e  htJ . /M´  :  which implies htJ . / has to bounded. Therefore: P .a; C / D f 2  C  W htJ . / is boundedg  So we can partition P .a; C / into finitely many subsets in each of which pj ; j … J are fixed while the rest of coefficients vary. However J is of finite type, hence each partition itself has to be finite, hence a 2 A[ . Proposition 8.29. A[ D E XP.I NT.T //. Proof. Using Lemma 8.27 and 8.28 we have: E XP.I NT.T // D E XP.W Dfin / A[ . This combined with Lemma 8.26 shows E XP.I NT.T // D A[ .  74  € MO 8.4. A Subset of G  8.4.2  Decay and Minima  € [ ´ N C A[ K, then the statement of Theorem 8.23 can be Definition 8.30. Set G €[ € MO . Note that this is a strict inclusion since 1 2 G € MO but rewritten as G G €[. 1…G Definition 8.31. For a 2 AC and C > 0, let P .a; C / be a subset of P defined as follows: if  2 P .a; C / then  and all weights in P of lower depth belong to  P .a; C /. Definition 8.32. For a 2 AC and C > 0 define: M  L. I a; C / ´  L. / ;  2P .a; C /  € .a; C / ´  ˚  1  2€ W  « 1 2 L. I a; C / :  Remark 8.33. L. I a; C / is a N C AC -stable subspace of L. / for all a 2 AC and C > 0. Lemma 8.34. If a 2 A[ then for all C > 0, L. I a; C / is finite dimensional. Proof. a 2 A[ so P .a; C / is finite for all C > 0. From the structure of the highest weight module it follows that P .a; C / is finite for all C > 0 as well. Since the weight spaces of L. / are finite dimensional, L. I a; C / is finite dimensional for all C > 0. € [ , for all C > 0; ˚ achieves a minimum on the set: Lemma 8.35. If g 2 G € .gA ; C /g. € [ , since ˚ is right invariant under K we may assume that g 2 Proof. Let g 2 G N C AC . Take C > 0 to be arbitrary and consider the following set: SD  ˚  1  1 W  2 € .gA ; C /  «  L. /Z \ L. I gA ; C /:  To prove the Lemma we need to show that g length. However L. I gA ; C / is  N C AC -stable  1 .S /  has an element of minimal  and therefore g  1 .S /  still lies in 75  € MO 8.4. A Subset of G € [ , by Lemma 8.34 L. I gA ; C / is finite dimensional. L. I gA ; C /. Since g 2 G Therefore g  1 .S /,  as a discrete subset of the finite dimensional subspace, has an  element of minimal length. € v 2 L. /Z and C > 0. If v … L. I gA ; C / then: Lemma 8.36. Let g 2 G; g  1  v >C  1.  Proof. Since v … L. I gA ; C /, based on the construction of P .gA ; C / we see that among all the weights of maximal depth that appear in the weight decompo… P .gA ; C /. Therefore  sition of v, there is at least one weight, , such that … P .gA ; C / as well and we have: gA < C:  (8.37)  On the other hand Lemma 8.4 gives us: g Since v 2 L. /Z we have: jc j  1  gA jc j :  v  1, which gives us: g  1  v  gA :  (8.38)  Combining (8.38) and (8.37) gives the desired result. € Corollary 8.39. Let g 2 G; C  2 € and C > 0. If  … €.gA ; C / then: ˚. g/ >  1.  € Corollary 8.40. Let g 2 G;  2 € . Then  2 € gA ; ˚. g/  1  .  € [ , by Lemma 8.35 ˚ achieves its minimum in € .gA ; C / for all Let g 2 G C > 0. So pick an arbitrary element  2 € and let € gA ; ˚.Âg/  1  . We claim  0  0  denote the minimum in  is the minimum on the entire orbit: € g. Suppose  2 € is arbitrary, there are two cases: 2 € gA ; ˚.Âg/  1  : then ˚. 0 g/ Ä ˚. g/ follows from our choice of  0.  76  € [ is not € -invariant 8.5. G … € gA ; ˚.Âg/  1  : then ˚.Âg/ < ˚. g/ holds by Lemma 8.39. But  from Corollary 8.40 we have  2 € gA ; ˚.Âg/  1  and therefore: ˚. 0 g/ Ä  ˚.Âg/. € MO . This comThus we have a global minimum on € g and therefore g 2 G pletes the proof of Theorem 8.23.  8.5  € [ is not € -invariant G  € [ is not in fact In this section we will show using indefinite GCMs of rank 2 that G € -invariant. But first we need to carry on a few calculations in SL2 .  8.5.1  Iwasawa Decomposition in SL2  Definition 8.41. For u 2 R and t 2 R>0 define: .u/ D  !  1 u 1  Á.t / D  ;  !  t  ;  1  t  C .u/  D  1 u 1  ! :  Lemma 8.42. The Iwasawa decomposition of SL2 can be explicitly written as follows: a b c d where  D  p  !   D  C  ac C bd 2  à Á.  1  /  d  1  c  1  c d  1 1  ! ;  c2 C d 2.  Lemma 8.43. Let r be the non-trivial element of the Weyl group in SL2 : rD  1 1  ! :  Then its action on SL2 =SO 2 is given by: r  C .u/Á.t / D  C   à t u Á p : Á u2 C t 4 u2 C t 4  77  € [ is not € -invariant 8.5. G Proof. We use the Iwasawa decomposition: r  C .u/Á.t /  D  D  8.5.2  t  !  t  1  t  1u  0 à  t u Á @ Á p 2 u2 C t 4 u C t4  C  1  2 p t u2 Ct 4 A p u u2 Ct 4  p u u2 Ct 4 2 p t u2 Ct 4  The Counterexample  € be the group corresponding to the rank 2 GCM Let G ! n ; 2  2  A D Aij D  m  m; n 2 N; n  m:  € and consider it as an element of G=K D N C AC . € € [ is not € -invariant. Since the We will calculate r 1 na 2 G=K to show that G € and r 1 na 2 G KP we will minimal group, G KP (see §7.4) is a subgroup of G Let na D  1 .u/Á1 .t1 /Á2 .t2 /  do the calculations in the minimal group where we have concrete generators and relations for the whole group. First we introduce the following notation to simplify our task: D D  q  u2 C t14 ;  q  u2 C t14 t2 2m :  Now using Lemmas 8.42, 8.43 and 7.34 we have: r 1 na D r 1 ( D D D  1 .u/Á1 .t1 /Á2 .t2 /  uÁ 1  2  uÁ 1  2  uÁ 1  2  Á1  t1 Á  u  1  t12 (  1  t12  Á1 Á1  t1 Á Á2 .t2 /  t2  1  1  u  1  t12  1  t2m  1  u m  t12  Á2 .t2 /  1  u  t1 Á Á2 .t2 / Á2 .t2 /  ! )  1  1  t12  1  u  ) Á2 .t2 /  1  u t12  ! 1  1  !  1 1  78  8.6. Open Problems D  uÁ 1  2  (   1  D D D  uÁ 2  Á1 Á1  t1 Á  1  2  1  1  t12 2  à  Á1  t2m  t1 Á Á2 .t2 /  uÁ   D  t2 m  2  2  1  t1 Á Á2 .t2 /  ut12  uÁ 1  Á1   1  " 1  " t2  t2  m  ut12  2  u  2  t2 m  ut12  m  2  2 m ut1  à  t2  t2  t12 t2m  u t12 t2 m 1 à  à t2m Á1 #!  m t2 Á2 .t2 /Á1 #!  m  1  t2m  Á1  t1 Á Á2 .t2 /Á1  1  ! )  1 1  à    à  u  à  à t1 Á2 .t2 / 2 Á1  Now in this calculation lets consider what has happened at the level of AC -component. The initial point .t1 ; t2 / has been transformed into: ! t1 ; t2 : q u2 C t14 t2 2m First note that if one takes u D 0 then one recovers the simple action of the fundamental reflection on AC but if u ! 1 the first component approaches 0. Hence even if a 2 E XP.I NT.T // by taking u large enough in 1 .u/ 2 N˛1 we can make € [ is not € -invariant. sure that r1 na … N C E XP.I NT.T //. In other words G  8.6  Open Problems  € AR and G € MO . Question 8.44. Find a relationship between G € MO ). Question 8.45. Compute AAR and AMO (points in AC which also belong to G € AR and G € MO onto AC (This is a far Question 8.46. Compute the projection of G harder problem than computing AAR and AMO ). Question 8.47. Is N C =.N C \ € / a projective limit of finite dimensional compact spaces and hence compact?  79  Bibliography [1] Ling Bao, Lisa Carbone, and Howard Garland.  Integer forms of kac-  moody groups and eisenstein series in low dimensional supergravity theories. Preprint. [2] Nicole Bardy. Systèms de racines infinis. Mém. Soc. Math. Fr. (N.S.), (65):vi+188, 1996. [3] Armand Borel. Introduction aux groupes arithmétiques. Publications de l’Institut de Mathématique de l’Université de Strasbourg, XV. Actualités Scientifiques et Industrielles, No. 1341. Hermann, Paris, 1969. [4] Nicolas Bourbaki. Lie groups and Lie algebras. Chapters 7–9. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 2005. Translated from the 1975 and 1982 French originals by Andrew Pressley. [5] Lisa Carbone and Howard Garland. Infinite dimensional chevalley groups and kac-moody groups over Z. Preprint. [6] Lisa Carbone and Howard Garland. Existence of lattices in kac-moody groups over finite fields. Commun. Contemp. Math, 5(5):813–867, 2003. [7] R. W. Carter. Lie algebras of finite and affine type, volume 96 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2005. [8] Tom De Medts, Ralf Gramlich, and Max Horn. Iwasawa decompositions of split kac-moody groups. Journal of Lie Theory, 19(2):311–337, 2009.  80  Bibliography [9] Jacques Dixmier. Enveloping algebras. North-Holland Publishing Co., Amsterdam, 1977. North-Holland Mathematical Library, Vol. 14, Translated from the French. [10] H. Garland. Certain Eisenstein series on loop groups: convergence and the constant term. In Algebraic groups and arithmetic, pages 275–319. Tata Inst. Fund. Res., Mumbai, 2004. [11] Howard Garland.  The arithmetic theory of loop algebras.  J. Algebra,  53(2):480–551, 1978. [12] Howard Garland. The arithmetic theory of loop groups. Inst. Hautes Études Sci. Publ. Math., (52):5–136, 1980. [13] Howard Garland. Eisenstein series on loop groups: Maass-Selberg relations. I. In Algebraic groups and homogeneous spaces, Tata Inst. Fund. Res. Stud. Math., pages 275–300. Tata Inst. Fund. Res., Mumbai, 2007. [14] V. G. Kac and D. H. Peterson.  Defining relations of certain infinite-  dimensional groups. Astérisque, (Numero Hors Serie):165–208, 1985. The mathematical heritage of Élie Cartan (Lyon, 1984). [15] Victor G. Kac. Infinite-dimensional Lie algebras. Cambridge University Press, Cambridge, third edition, 1990. [16] Shrawan Kumar. Kac-Moody groups, their flag varieties and representation theory, volume 204 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 2002. [17] Olivier Mathieu. Formules de caractères pour les algèbres de Kac-Moody générales. Astérisque, (159-160):267, 1988. [18] Robert V. Moody and Arturo Pianzola. Lie algebras with triangular decompositions. Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley & Sons Inc., New York, 1995. A Wiley-Interscience Publication.  81  Bibliography [19] D. H. Peterson and V. G. Kac. Infinite flag varieties and conjugacy theorems. Proc. Natl. Acad. Sci. USA, 80(6):1778–1782, 1983. [20] Guy Rousseau. Groupes de kac-moody déployés sur un corps local ii: Masures ordonnées. In Preparation, 2012. [21] Jean-Pierre Serre. Trees. Springer-Verlag, Berlin, 1980. Translated from the French by John Stillwell. [22] N. N. Shapovalov. On a bilinear form on the universal enveloping algebra of a complex semisimple lie algebra. Functional Analysis and Its Applications, 6:307–312, 1972. 10.1007/BF01077650. [23] Peter Slodowy. Singularitäten Kac-Moody-Lie algebren, assoziierte Gruppen und Verallgemeinerungen. 1984. Habilitationsschrift–Universität zu Bonn. [24] Robert Steinberg. Lectures on Chevalley groups. Yale University, New Haven, Conn., 1968. Notes prepared by John Faulkner and Robert Wilson. [25] Jacques Tits. Définition par générateurs et relations de groups avec BN paires. C. R. Acad. Sci. Paris Sér. I Math., 293(6):317–322, 1981. [26] Jacques Tits. Resume de cours. Annuaire College de France, 81:75–87, 1981. [27] Jacques Tits. Resume de cours. Annuaire College de France, 82:91–106, 1982. [28] Jacques Tits. Uniqueness and presentation of Kac-Moody groups over fields. J. Algebra, 105(2):542–573, 1987. [29] Zhe Xian Wan. Introduction to Kac-Moody algebra. World Scientific Publishing Co. Inc., Teaneck, NJ, 1991.  82  Index of Notation L. /C , 19  P , 20  L. /R , 22  P .a; C /, 73  L. I a; C /, 75  ˘, 4  M. /C , 18  ˘ _, 4  C , 19  Q, 7  D, 16  Q , 40  Dfin , 16  Q_ , 7  ,7  S, 20  im ,  15  S , 21  im ˙ , 15 re , 15  T , 16  re ˙,  15  XQ .m/, 35  ˙,  8  YQ .m/, 50  UZ .m/, 47  FJ , 16  ˛i , 4  HF.m/, 45  ˛i_ , 4  HQ .m/, 35  ˙ .u/,  77  HZ .m/, 47  dp, 20  € , 52  Á.t /, 77  € .a; C /, 75 € AR , 66 G  EŒ , 22  € [ , 75 G € MO , 70 G  ht, 7  € , 70 G €F, 45 G  aC , 8 aR , 15  €Q , 45 G  aZ , 28  P, 33  bC , 8  Pi , 33  b˙i;Q , 40  Á, 20 htJ , 74  83  Index of Notation ci;Q , 40  P .a; C /, 75  gC , 5  , 65  gR , 22  e˙i , 5  gZ , 28  ri , 13  gi;F, 62  riad , 12  ˙ nC ,8  ri , 12  ˙ nZ ,  28  ni;Q , 40 pi;C , 8 , 20 0 , 22 AC F , 61  AAR , 67 A[ , 73 A , 70 Ai , 71 AQ , 40 BQ , 40 B˙i;Q , 40 Ci;Q , 40 KP GF , 63  Gi;F, 62 K, 60 Ki , 60 Li;Q , 40 ˙ NQ , 40  N˙˛i ;Q , 40 Ni;Q , 40 Pi;Q , 40 W , 13 , 70 !, 5 1 , 18 84  Appendix A  Formulas in Associative Algebras Let F be a field of characteristic zero. In this appendix we present a number of formulas that hold in any associative F-algebra AF with a unit. Notation A.1. For x 2 AF and n 2 N we set: xn x Œn D ; nŠ ! x x.x D n  1/  .x nŠ  n C 1/  :  Also we may denote ad.x/.y/ by Œx; y when the latter is more convenient. Lemma A.2 ([4] page 178). If x; y 2 AF and n 2 N: X 1 ad.x/n .y/ D . 1/q x Œp yx Œq nŠ pCqDn  Lemma A.3 ([4] page 178). Suppose ´; x 2 AF such that Œ´; x D cx for c 2 F. Then for all n 2 N, and all P 2 FŒX , we have: P .´/x Œn D x Œn P .´ C nc/: Definition A.4. .x; y; ´/ with x; y; ´ 2 AF is called a S-triple if: Œx; y D ´;  Œ´; x D 2x;  Œ´; y D  2y:  Lemma A.5 ([4] page 70). If .x; y; ´/ is a S -triple in AF then we have the follow-  85  Appendix A. Formulas in Associative Algebras ing: ´; x n D 2nx n ´; y n D  2ny n  y; x n D nx n  1  . ´  x; y n D ny n  1  .´  1/x n  n C 1/ D n. ´ C n n C 1/ D n.´ C n  1/y n  1  1  Lemma A.6 ([24] page 9). If .x; y; ´/ is a S -triple in AF for p; q 2 N, we have: x Œp y Œq D  min.p;q/ X rD0  y Œq  r  ´  p C 2r  q r  ! x Œp  r  (A.7)  86  Appendix B  Lie Algebras Let R be a commutative ring with a unit element. Definition B.1. An R-module LR , is called an R-algebra if one has a R-homomorphism: W LR ˝R LR ! LR . op  Definition B.2. Given an R-algebra LR we define its opposite algebra, LR to be op .x  the identical to LR as an R-module but with  ˝ y/ D  .y ˝ x/.  Definition B.3. An R-linear map D W LR ! LR is called a derivation if it satisfies Leibniz’s law: D.xy/ D D.x/y C xD.y/. Lemma B.4. If D is a derivation and x1 ; x2 ; D.x1 x2  xk / D  k X  x1  xi  ; xk 2 LR then: 1 D.xi /xi C1  xk :  i D1  Definition B.5. For x 2 LR define ad.x/ W LR ! LR by ad.x/.y/ D xy  yx.  Lemma B.6. ad.x/ is a derivation for all x 2 LR . Definition B.7. A R-Lie algebra is an R-algebra with the following properties: 1) The map  W LR ˝R LR ! LR admits a factorization: LR ˝R LR !  2 ^  LR ! LR ;  lets denote the image of x ˝ y under this map by Œx; y then condition becomes: Œx; x D 0;  8x 2 LR :  87  Appendix B. Lie Algebras 2) We have Jacobi’s identity: ŒŒx; y; ´ C ŒŒy; ´; x C ŒŒ´; x; y D 0 Definition B.8. Given a R-Lie algebra LR we define its opposite Lie algebra with the following bracket: Œx; yop ´  Œx; y:  0 0 is called a be two R-Lie algebras, a map ' W LR ! LR Definition B.9. Let LR ; LR  Lie homomorphism if ' is R-linear and satisfies: ' .Œx; y/ D Œ'.x/; '.y/, for all x; y 2 LR . Example B.10. Let LR be an arbitrary R-algebra. One can equip LR with a Lie algebra structure by defining: Œx; y D 0 for all x; y 2 LR . Such a Lie algebra is called commutative. Example B.11. The set D ER.LR / of all derivations of an R-algebra LR is a Lie algebra with the product: ŒD; D 0  D DD 0  D 0 D.  Example B.12. Let LR be an R-algebra then Œx; y D xy  yx turns LR into an  R-Lie algebra. We use L IE.LR / when want to emphasize the Lie algebra structure on LR . Note that the underlying sets of LR and L IE.LR / are identical. Theorem B.13. Let LR be an R-Lie algebra. For any x 2 LR define a map ad.x/ W LR ! LR by ad.x/.y/ D Œx; y, then: 1) ad.x/ is a derivation of LR . 2) The map x 7! ad.x/ is a Lie homomorphism of LR into D ER.LR /. Definition B.14. A universal enveloping algebra of an R-Lie algebra LR is a pair ."; UR /, where: UR is an associative R-algebra with a unit. " W LR ! UR is a Lie algebra homomorphism. H OMR  lie .LR ; L IE .TR //  Š H OMR  alg .UR ; TR /.  88  Appendix B. Lie Algebras Any Lie algebra possesses a universal enveloping algebra, moreover we have the following functorial properties: UR .L ˚ L0 / D UR .L/ ˝R UR .L0 /: 0 then by universal If we have a R-Lie algebra homomorphism: « W LR ! LR  property: U.« / W UR .L/ ! UR .L0 /: is a homomorphism of R-algebras. PBW Theorem ( [24] page 8). Let LR be an R-Lie algebra and ."; UR / its universal enveloping algebra. Then: " is injective. If LR is identified with its image in UR and if fx1 ; x2 ; R-basis for LR , then all monomials ai  ai  xi1 1 xi2 2 in which i1 <  < ik and ai1 ;  g is an ordered  ai  x ik k ;  ; aik 2 N, form an R-basis for UR .  89  Appendix C  Hopf Algebra C.1  Definition  Let R be a commutative ring with a unit. A Hopf algebra over R is a 6-tuple, .HR ; ; ; ; ; / such that: HR is an R-algebra with ; defining product and unit. HR is an R-coalgebra with ; defining co-product and co-unit. These two structures are compatible, i.e. ; are R-algebra homomorphisms. W HR  ! HR is an R-algebra homomorphism, usually called antipode,  such that the following diagram commutes: HR9 ˝R HR  1˝  / HR ˝R HR % / HR 9  /R  HR %  HR ˝R HR  ˝1  / HR ˝R HR  Given a Hopf algebra HR over R we can equip H OMR  alg .HR ;  R/ with the structure  of a group scheme over R.  90  C.2. Enveloping Algebra  C.2  Enveloping Algebra  Let gR be a R-Lie algebra and consider the following Lie algebra homomorphism:  ˚  gR ! f0g  ˚  ;  x 7! 0  gR ! gR ˚ gR  ˚  ;  x 7! .x; x/  op  gR ! gR x 7!  :  x  Then we have the corresponding maps for enveloping algebras: W UR .g/ ! UR .f0g/ D R; W UR .g/ ! UR .g ˚ g/ D UR .g/ ˝R UR .g/; W UR .g/ ! UR .gop / D UR .g/op : Consider gR as a subset of UR .g/ then for x 2 gR then: .x/ D 0; .x/ D 1 ˝ x C x ˝ 1 and .x/ D  x.  Definition C.1. An element x 2 UR .g/ is called primitive if: .x/ D x ˝1C1˝x. Lemma C.2. UR .g/; ; ; ; ;  is a co-commutative Hopf algebra, where:  is the multiplication, is the unit, is the co-multiplication, is the co-unit, is the antipode.  C.3  The Dual of the Enveloping Algebra  The material in the section follows [9] §2.7.8. Consider the transpose of the co-product: W UR .g/ ˝R UR .g/  ! UR .g/  91  C.3. The Dual of the Enveloping Algebra combined with the restriction: UR .g/ ˝R UR .g/  UR .g/ ˝R UR .g/  W UR .g/ ˝R UR .g/ ! UR .g/ . More explicitly for  we get a linear map:  f; g 2 UR .g/ the product is give by: .fg/.u/ D  .f ˝ g/.u/ D .f ˝ g/. .u//:  The vector space UR .g/ is thus equipped with the structure of an algebra. Notation C.3. Let Z1 C be the set of all infinite sequences of non-negative integers and for  2 Z1 C define: 1 Y  x D e D  nD1 1 Y  xnn 2 RŒŒx1 ; x2 ;    enŒ n  2 UR .g/:  nD1  Lemma C.4. For  2 Z1 C we have: .e / D  X  e ˝e :  C D  Proof. We have: Œ  e1 1    Á  D D D  1 1Š  1 1Š  e1 1 D  1Š  .e1 /  .e1 ˝ 1 C 1 ˝ e1 /  X 1C  1  1D 1  1  1  1 e1 1 ˝ e1 1 Š Š 1 1  Hence: .e / D  n Y  Œ  ei i  !  i D1  92  C.3. The Dual of the Enveloping Algebra  D D  n Y  Œ i  ei  i D1 n Y i D1  D  D  Á  1 ei 1 ˝ ei Š i iŠ  X iC  iD i  1  X 1C  1D 1  nC  nD n  X  1Š  1Š  nŠ  nŠ  1  e1 1  en n ˝ en 1  en n  e ˝e  C D  Theorem C.5. There is an isomorphism of the R-algebras: UR .g/ Š RŒŒx1 ; x2 ;    given by the map: f 7! sf ´ ˚ Proof. Since e W  X  f .e /x :  « 2 Z1 C is a basis for UR .g/ the mapping f 7! sf is bijective  (in the same way that the dual of an infinite direct sum is an infinite direct product). Moreover if f; g 2 UR .g/ , then: sfg D  X  .fg/.e /  2Z1 C  D  X  .f ˝ g/. .e //  2Z1 C  0 D  X  .f ˝ g/ @  2Z1 C  D D  1 X  e ˝e A  C D  X  X  2Z1 C  C D  X  f .e / ˝ g.e /  f .e /g.e /x  C  D sf sg :  ; 2Z1 C  In particular, the algebra UR .g/ is associative and commutative. Its unity, 1 , is a linear form such that Ker.1 / D AR .g/ and 1 .1/ D 1. The unit map then is the transpose co-unit map in UR .g/, that is we have:  W R ! UR .g/ , where  93  C.3. The Dual of the Enveloping Algebra .1/ D 1 . Moreover the transpose of the principal anti-automorphism of UR .g/ is an automorphism of UR .g/ and is called the principal automorphism of UR .g/ .  94  Appendix D  Amalgams and Tits Systems The material of this appendix follows [21].  D.1  Direct Limits  Let .Gi /i 2I be a family of groups such that for each pair we have a homomorphism fij W Gi ! Gj . A group G and a family of homomorphisms .gi W Gi ! G/i 2I is called the direct limit of the family .Gi /i 2I relative to .fij / if: (1) gj ı fij D gi for all i; j 2 I . (2) If H is a group with a family of homomorphisms .hi W Gi ! H /i 2I such that hj ı fij D hi for all i; j 2 I . Then there is exactly one homomorphism W G ! H such that hi D  ı gi  Proposition D.1. The pair consisting of G and the family of homomorphisms .gi W Gi ! G/i 2I exists and is unique up to unique isomorphism.  D.2  Tits Systems  A Tits System is a 4-tuple .G; B; N; S / where G is a group, B and N are subgroups of G, and S a subset of W D N=.B \ N / satisfying the following axioms: (1) The set B [ N generates G and B \ N is a normal subgroup of N . (2) The set S generates W D N=.B \ N / and consists of elements of order 2. (3) BsB BwB  BwB [ BswB for s 2 S and w 2 W .  (4) For each s 2 S one has sBs  1  ¤ B. 95  D.3. Tits’s Theorem The group W is called the Weyl group of .G; B; N; S /; the pair .W; S / is a Coxeter system, that is S and the relations: .st /mst D 1;  s; t 2 S and mst < 1;  is a presentation for W . The group G is the disjoint union of double cosets BwB; w 2 W . This is called the Bruhat decomposition. If S 0  S , let WS 0 be the subgroup of W generated by elements of S 0 ; then  PS 0 D BWS 0 B. Then S 0 7! PS 0 is a bijection of the set of subsets of S onto the set of subgroups of G containing B. PS 0 is then called the standard Parabolic subgroup of type S 0 .  D.3  Tits’s Theorem  Let G be a group, and let .Gi /i2I be a family of subgroups of G. We say G is the product of Gi amalgamated along their intersections if G is the direct limit of the system formed by the Gi , the Gi \ Gj and the inclusions: Gi \ Gj  Gi ;  Gi \ Gj  Gj :  Theorem D.2 ([25]). Let .G; B; N; S / be a Tits system; Then G is the product of N and .Pfsg /s2S amalgamated along their intersections.  96  

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