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Shifted q = 0 affine algebras Hsu, You-Hung 2019

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Shifted q = 0 affine algebrasbyYou-Hung HsuA THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Mathematics)The University of British Columbia(Vancouver)May 2019c© You-Hung Hsu, 2019The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoc-toral Studies for acceptance, the dissertation entitled:Shifted q = 0 affine algebrassubmitted by You-Hung Hsu in partial fulfillment of the requirements forthe degree of Doctor of Philosophyin MathematicsExamining Committee:Sabin CautisSupervisorKai BehrendSupervisory Committee MemberBen WilliamsSupervisory Committee MemberKaru KalleUniversity ExaminerMark Van RaamsdonkUniversity ExamineriiAbstractIn this thesis, we accomplish the following three things.1. Defining the shifted q = 0 affine algebras (Chapter 2).2. Explaining what it means for such algebra to act on categories (Chapter 3).3. Giving an example of such a categorical action (Chapter 5).Our motivation comes from the categorification of quantum groups and their action on categories. On thederived categories of coherent sheaves on Grassmannians or partial flag varieties, we try to understand an actionvia using the language of Fourier-Mukai transformations with kernels inducing by natural correspondences.After decategorifying, the q = 0 shifted affine algebras are similar to the shifted quantum affine algebrasdefined by Finkelberg-Tsymbaliuk [FT], where some of the relations can be obtained from their relations bytaking v = 0 (i.e. the q-analogue).Finally, we relate shifted q = 0 affine algebras to q = 0 affine Hecke algebras. In particular, we use theaction of shifted q = 0 affine algebras to construct an action of the q = 0 affine Hecke algebras on the derivedcategory of coherent sheaves on full flag varieties. We also relate this action to the notion of Demazure descentwhich was introduced in [AK1].iiiLay SummaryIn mathematics we study symmetries of objects. For example, we might study algebras acting on vector spacesor more generally on categories.One elementary example involves the Lie algebra sl2(C). A representation of sl2(C) is a vector space Vwhere the elements of sl2(C) act as operators on V . This is equivalent to the following data....e,,Vλ−2fjje**Vλe,,fll Vλ+2fjje))....fllsuch that [e, f ]|Vλ = λIdVλ .Categorification involves replacing vector spaces with categories and looking for extra structure in the pro-cess. One way to categorify some representations of sl2(C) is using the categories of constructible sheaves onGrassmannians G(k,N).In this thesis, we replace constructible sheaves with coherent sheaves on G(k,N). Roughly speaking, con-structible sheaves relate to the study of locally constant functions while coherent sheaves relate to the study ofholomorphic functions. We call the resulting new algebra the shifted q = 0 affine algebra and describe how itacts on these categories of coherent sheaves.ivPrefaceAll of the work presented in this thesis was conducted as I was a Ph.D. student in the Mathematics Departmentof University of British Columbia.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Categorical g action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Categorical action on derived categories of coherent sheaves . . . . . . . . . . . . . . . . . . . 21.3 q = 0 affine Hecke algebra and Demazure descent . . . . . . . . . . . . . . . . . . . . . . . . . 52 The definition of shifted q = 0 affine algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1 Shifted quantum affine algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Shifted q = 0 affine algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Categorical action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Preliminaries on coherent sheaves and Fourier-Mukai kernels . . . . . . . . . . . . . . . . . . . 155 A geometric example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.1 Correspondence and related varieties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.2 sl2 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.3 The rest relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49vi6 q = 0 affine Hecke algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.1 Definition and action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.2 Relation to Demazure descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62A Another presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69viiAcknowledgmentsI would like to thank my supervisor Professor Sabin Cautis, not only for his guidance and motivation during myPh.D. study, but also for his support during my tough times. I would also like to thank Professor Kai Behrendand Professor Ben Williams for serving on my thesis committee. I would also like to thank my parents and myfriends for their encouragement and support.viiiChapter 1IntroductionWe study categorical actions on the bounded derived categories of coherent sheaves on Grassmannians andpartial flag varieties. This thesis is divided into three parts. In the first part, we define the shifted q = 0 affinealgebras and give a definition of what it means to have a categorical action (Chapter 2 and 3).In the second part, we prove that there is a categorical action of shifted q = 0 affine algebra on the boundedderived categories of coherent sheaves of Grassmannians and partial flag varieties (Chapter 4 and 5).In the third part, we relate shifted q = 0 affine algebras to q = 0 affine Hecke algebras (Chapter 6). We alsoconstruct an action on the bounded derived categories of coherent sheaves on full flag variety and relate it to thenotion of Demazure descent.1.1 Categorical g actionDuring the past decade, there has been much progress on categorical actions of semisimple (or more generallyKac-Moody) Lie algebras g and their q-analogues. We refer to [CR], [KL1], [KL2], [KL3], and [R] for details.Considering the simplest case g = sl2(C). We would like to focus on geometric constructions of suchactions. Based on the work of [BLM], the derived category of constructible sheaves on GrassmanniansG(k,N),denoted by Dbc(G(k,N)), is a natural choice to construct categorical sl2-action. This means that the weightspaces Vλ are replaced by categories C(λ) = Dbc(G(k,N)), where λ = N − 2k.The linear map e : Vλ → Vλ+2 would be replaced by functor E : Dbc(G(k,N)) → Dbc(G(k − 1, N)). The1construction of E is via the following correspondence diagramFl(k − 1, k) = {0 k−1⊂ V ′ 1⊂ V N−k⊂ CN}p1ttp2**G(k,N) G(k − 1, N)where Fl(k − 1, k) is the 3-step partial flag variety. Then we define E := p2∗p∗1. The construction for F issimilar.So the representation of sl2 at the categorical level becomes....E00 Dbc(G(k + 1, N))FkkE.. Dbc(G(k,N))E ..Fnn Dbc(G(k − 1, N))FnnE++ ....Fppand we have the following theoremTheorem 1.1.1 (Beilinson-Lusztig-MacPherson, Rouquier). This gives a categorical sl2 action. More precisely,we haveEF|C(λ) ∼= FE|C(λ)⊕Id⊕λC(λ) if λ ≥ 0FE|C(λ) ∼= EF|C(λ)⊕Id⊕−λC(λ) if λ ≤ 0Using the theory ofD-modules, these relate to the derived category of coherent sheaves on cotangent bundleof Grassmannians. In [CKL2], the authors introduced the notion of geometric categorical sl2-action via usingthe language of Fourier-Mukai kernels on bounded derived categories of coherent sheaves, which is closelyrelated to the notion of a strong categorical sl2-action due to [CR]. They constructed geometric categoricalsl2-action on the bounded derived category of coherent sheaves on the cotangent bundle of Grassmannians, andalso prove that such action gives derived equivalences between these categories, see [CKL3] for details. Laterthey generalized it to geometric categorical g-action and used it to construct a braid group action [CK1].1.2 Categorical action on derived categories of coherent sheavesInspired by the above work, we consider coherent sheaves instead of constructible sheaves. This leads us toconstruct actions of the shifted q = 0 affine algebra on the bounded derived categories of coherent sheaves onG(k,N) or Flk(CN ).After decategorifying, we obtain an algebra where the relations look similar to the relations in the second2presentation of the shifted quantum affine algebra defined by [FT]. But there is a variable v (i.e. the q-analogue)in their definition which comes from the C∗-action. In our case, the natural C∗-action on Grassmannians istrivial, so we do not have a variable like q. Since some of our relations can be obtained from their relations bytaking v = 0, we call the algebra shifted q = 0 affine algebra.Let us explain more details about our result. The higher representation theory of Kac-Moody Lie algebrag involves the action of Uq(g) on categories. This means that to each weight λ of g one assigns an additive(graded) category C(λ) and to generators ei and fi of Uq(g) one assigns functors Ei : C(λ) → C(λ + αi) andFi : C(λ+αi)→ C(λ). These functors are then required to satisfy certain relations analogous to those in Uq(g).For example, the relation [ei, fi] =Ki−K−1iq−q−1 becomesEiFi|C(λ) ∼= FiEi|C(λ)⊕ ⊕[〈λ,αi〉]IdC(λ) if 〈λ, αi〉 ≥ 0 (1.2.1)and an analogous relation for the condition 〈λ, αi〉 ≤ 0. Here [n] := q−n+1 + q−n+3 + ....+ qn−3 + qn−1 is thequantum integer and⊕[〈λ,αi〉](.) denotes a graded direct sum.One can extend such an action to affine Kac-Moody Lie algebras ĝ with main generators ei,r, fi,r wherer ∈ Z. Usually, the extra degree r ∈ Z would come from twisting by a natural line bundle. Following [CL],such an action consists of a target category which is a graded triangulated C-linear 2-category with some 1-morphisms Ei,r1λ, 1λFi,r, r ∈ Z.For its categorical action, as an example, the commutator relation [ei,r, fi,s]1λ = −q−r−1−λiPi1λ, wherer + s = −1 in the categorical level is given by the following exact triangle→ Fi,sEi,r1λ → Ei,rFi,s1λ → Pi1λ〈−λi − r − 1〉[1]→ if r + s = −1 (1.2.2)where λi = 〈λ, αi〉 and 〈1〉 denote the grading shift. Note that usually the exact triangles are non-split.For the shifted q = 0 affine algebra, we consider the case where g = sln. Then we can write theweight λ as k = (k1, ..., kn). Now we lift the weight spaces to weight categories, which are the boundedderived categories of coherent sheaves on Grassmannians or partial flag varieties. Similarly, we have generatorsei,r1k, fi,s1k, (ψ±i )±11k, hi,±11k that lift to 1-morphisms Ei,r1k, Fi,s1k, (Ψ±i )±11k, Hi,±11k. Many of thecategorical relations between those 1-morphisms are described by (non-split) exact triangles. For example, thecommutator relation [ei,r, fi,s]1k = ψ+i 1k, where r+ s = ki+1 in the categorical level is given by the followingexact triangle→ Fi,sEi,r1k → Ei,rFi,s1k → Ψ+i 1k →, if r + s = ki+1 (1.2.3)3which is similar to (1.2.2). For other categorical relations we refer to Definition 3.0.1 for details. Again, theexact triangle here is non-split.Also, by doing categorification, we find interesting hidden higher relations that are trivial on the level of K-theory but non-trivial on the level of derived categories. For example, the relation between the generators hi,11kand ei,r1k is [hi,±1, ej,r]1k = 0. But their relation in the categorical level is given by the following (non-split)exact triangle→ Hi,1Ei,r1k → Ei,rHi,11k → (Ei,r+1⊕Ei,r+1[1])1k → (1.2.4)which implies that the relation is NOT trivial on derived categories.As an example, we consider the special case where k = (k, 2 − k) = (2, 0), (1, 1), (0, 2), and the corre-sponding weight categories are the derived categories of coherent sheaves on Grassmannians.Db(G(2, 2)) Db(G(1, 2) = P1) Db(G(0, 2)).EFEFWe drop the subscript i to simplify notations. Since we are using the geometric categorical action, thefunctors Er1(k,2−k) and H11(k,2−k) are determined by Fourier-Mukai kernels, and we denote them by Er1(k,2−k)and H11(k,2−k), respectively. Denoting ∗ to be the convolution of Fourier-Mukai kernels, the convolution ofFourier-Mukai kernels is again an objects in the derived categories. So the categorical relations would becomeexact triangles that relate Fourier-Mukai kernels.Consider P1 × P1 = {(V, V ′′)| dimV = dimV ′ = 1}, and let V , V ′′ to be the tautological line bundles onP1 × P1 and C2 to be the trivial bundle on P1. Denote ∆ ⊂ P1 × P1 to be the diagonal which is also a divisor.Then the kernelH11(1,1) is O2∆ ⊗ C2/V , which is determined by the following exact triangle in Db(P1 × P1)→ O∆ ⊗ V → H11(1,1) = O2∆ ⊗ C2/V → O∆ ⊗ C2/V → .The kernel E1(1,1) is OP1 , so after doing convolution with E1(1,1), we get the following exact triangle inDb(P1)→ V → (E ∗ H1)1(1,1) → C2/V → .By doing some computations, we show that this exact triangle splits, so (E ∗ H1)1(1,1) = V ⊕ C2/V .Similarly, we obtain the following exact triangle in Db(P1)→ 0→ (H1 ∗ E)1(1,1) = C2 → C2 → .4Combining them together, we obtain the following exact triangle→ (H1 ∗ E)1(1,1) = C2 → (E ∗ H1)1(1,1) = V ⊕ C2/V → V ⊕ V[1]→ (1.2.5)where V is the kernel E11(1,1). Note that this agrees with (1.2.4).There is much more for further study. For example, we would like to understand higher relations, i.e.their natural transformations. For instance, one might wonder about End(Ei,rEi,s) or End(Ei,rEj,s) or any othercomposition about Ei,r and Fj,s, like Khovanov-Lauda and Chuang-Rouquier did in [CR], [KL1], [KL2], [KL3].1.3 q = 0 affine Hecke algebra and Demazure descentSimilar like quantum affine algebras are related to affine Hecke algebras, we relate shifted q = 0 affine algebrasto q = 0 affine Hecke algebras in Chapter 6. As an application, we use the relations that we obtain from thecategorical action of shifted q = 0 affine algebras to construct categorical actions of q = 0 affine Hecke algebrason the bounded derived categories of coherent sheaves on the full flag varieties Db(Fl).On the other hand, there is a notion called Demazure descent defined in [AK1]. We explain how this notionrelate to the categorical action of q = 0 affine Hecke algebras.5Chapter 2The definition of shifted q = 0 affine algebras2.1 Shifted quantum affine algebrasIn this section, we give the definition of shifted quantum affine algebras from [FT]. Note that there are twodefinitions in loc. cit., we will chose the simple one, i.e. the Levedorskii type presentation.Let g be a simple Lie algebra, h ⊂ g be a Cartan subalgebra and (·, ·) be a non-degenerated invariantsymmetric bilinear form on g. Let {α∨i }i∈I ⊂ h∗ be the simple positive roots of g relative to h and cij :=2(α∨i ,α∨j )(α∨i ,α∨i ), di =(α∨i ,α∨i )2 for any i, j ∈ I . We fix the notations qi := qdi , and [m]q := qm−q−mq−q−1 .Definition 2.1.1. Given antidominant coweights µ1, µ2, set µ = µ1+µ2. Define b1,i := α∨i (µ1), b2,i := α∨i (µ2),bi = b1,i + b2,i. Then we define the shifted quantum affine algebra, denoted by Uˆµ1,µ2 , to be the associated C(q)algebra generated by{ei,r, fi,s, (ψ+i,0)±1, (ψ−i,bi)±1, hi,±1|i ∈ I, b2,i − 1 ≤ r ≤ 0, b1,i ≤ s ≤ 1}subject to the following relations{(ψ+i,0)±1, (ψ−i,bi)±1, hi,±1}i∈I pairwise commute, (U1)(ψ+i,0)±1 · (ψ+i,0)∓1 = (ψ−i,bi)±1 · (ψ−i,bi)∓1 = 1 (U2)ei,r+1ej,s − qciji ei,rej,s+1 = qciji ej,sei,r+1 − ej,s+1ei,r (U3)qciji fi,r+1fj,s − fi,rfi,s+1 = fj,sfi,r+1 − qciji fj,s+1fi,r (U4)ψ+i,0ej,r = qciji ej,rψ+i,0, ψ−i,biej,r = q−ciji ej,rψ−i,bi, [hi,±1, ej,r] = [cij ]qiej,r±1, (U5)6ψ+i,0fj,s = q−ciji fj,sψ+i,0, ψ−i,bifj,s = qciji fj,sψ−i,bi, [hi,±1, fj,s] = −[cij ]qifj,s±1, (U6)[ei,r, fj,s] = 0 if i 6= j and [ei,r, fi,s] =ψ+i,0hi,1 if r + s = 1ψ+i,0−δbi,0ψ−i,biqi−q−1iif r + s = 00 if bi + 1 ≤ r + s ≤ −1−ψ−i,bi+δbi,0ψ−i,0qi−q−1iif r + s = biψ−i,bihi,−1 if r + s = bi − 1, (U7)1−cij∑r=0(−1)r[1− cijr]qieri,0ej,0e1−cij−ri,0 = 0,1−cij∑r=0(−1)r[1− cijr]qif ri,0fj,0f1−cij−ri,0 = 0, (U8)[hi,1, [fi,1, [hi,1, ei,0]]] = 0, [hi,−1, [ei,b2,i−1, [hi,−1, fi,b1,i ]]] = 0, (U9)for any i, j ∈ I and r, s such that the above relations make sense.Remark 2.1.2. The numbers cij form a matrix (cij) which is called the Cartan matrix. In particular, wheng = sln, we have cij =2 if i = j−1 if |i− j| = 10 if |i− j| ≥ 2, and di = 1 for all i. The Cartan matrix is gievn by(cij) =2 −1 0 . . . 0 0−1 2 −1 . . . 0 0.......... . .......0 0 0 . . . −1 2. This tells us the relations of the shifted quantum affine algebra for g = sln. For example, some of the relationsin (U5) are ψ+i,0ei,r = q2ei,rψ+i,0, [hi,±1, ei,r] = [2]qei,r±1. Similarly for other relations.2.2 Shifted q = 0 affine algebrasIn this section, we define the shifted q = 0 affine algebras. By imitating the article by [FT], we also have twoversions of our algebra. We will give the definition of the simple one first, which is similar to Definition 2.1.1.The second one will be given in the appendix ??.Unfortunately, we can not define the algebra without referring to weight spaces. This means that we replacethe identity by the direct sum of a system of projectors, one for each element of the weight lattices. They are7orthogonal idempotents for approximate the unit element. We refer to [Lu] for details of such modification.LetC(n,N) := {k = (k1, ..., kn) ∈ Nn|k1 + ...+ kn = N}.We regard each k as a weight for sln via the identification of the weight lattice of sln with the quotientZn/(1, 1, ..., 1). We choose the simple root αi to be (0...0,−1, 1, 0...0) where the −1 is in the i-th positionfor 1 ≤ i ≤ n − 1. Then we introduce the first version of shifted q = 0 affine algebra for sln, which is definedby using finite generators.Definition 2.2.1. Denote by U˙0,N (Lsln) the associative C-algebra generated byei,r1k, fi,s1k, (ψ+i )±11k, (ψ−i )±11k, hi,±11k, where 1 ≤ i ≤ n− 1, −ki − 1 ≤ r ≤ 0, 0 ≤ s ≤ ki+1 + 1with the following relations1k1l = δk,l1k, ei,r1k = 1k+αiei,r, fi,r1k = 1k−αifi,r, (ψ+i )±11k = 1k(ψ+i )±1, hi,±11k = 1khi,±1, (U01){(ψ+i )±11k, (ψ−i )±11k, hi,±11k|1 ≤ i ≤ n− 1, k ∈ C(n,N)} pairwise commute, (U02)(ψ+i )±1 · (ψ+i )∓11k = 1k = (ψ−i )±1 · (ψ−i )∓11k, (U03)ei,rej,s1k =−ei,s+1ei,r−11k if j = iei+1,sei,r1k − ei+1,s−1ei,r+11k if j = i+ 1ei,r+1ei−1,s−11k − ei−1,s−1ei,r+11k if j = i− 1ej,sei,r1k if |i− j| ≥ 2, (U04)fi,rfj,s1k =−fi,s−1fi,r+11k if j = ifi,r−1fi+1,s+11k − fi+1,s+1fi,r−11k if j = i+ 1fi−1,sfi,r1k − fi−1,s+1fi,r−11k if j = i− 1fj,sfi,r1k if |i− j| ≥ 2, (U05)8ψ+i ej,r1k =−ei,r+1ψ+i 1k if j = i−ei+1,r−1ψ+i 1k if j = i+ 1ei−1,rψ+i 1k if j = i− 1ej,rψ+i 1k if |i− j| ≥ 2, ψ−i ej,r1k =−ei,r+1ψ−i 1k if j = iei+1,rψ−i 1k if j = i+ 1−ei−1,r−1ψ−i 1k if j = i− 1ej,rψ−i 1k if |i− j| ≥ 2, (U06)ψ+i fj,r1k =−fi,r−1ψ+i 1k if j = i−fi+1,r+1ψ+i 1k if j = i+ 1fi−1,rψ+i 1k if j = i− 1fj,rψ+i 1k if |i− j| ≥ 2, ψ−i fj,r1k =−fi,r−1ψ−i 1k if j = ifi+1,rψ−i 1k if j = i+ 1−fi−1,r+1ψ−i 1k if j = i− 1fj,rψ−i 1k if |i− j| ≥ 2, (U07)[hi,±1, ej,r]1k =0 if i = j−ei+1,r±11k if j = i+ 1ei−1,r±11k if j = i− 10 if |i− j| ≥ 2, [hi,±1, fj,r]1k =0 if i = jfi+1,r±11k if j = i+ 1−fi−1,r±11k if j = i− 10 if |i− j| ≥ 2, (U08)[ei,r, fj,s]1k = 0 if i 6= j and [ei,r, fi,s]1k =ψ+i hi,11k if r + s = ki+1 + 1ψ+i 1k if r + s = ki+10 if − ki + 1 ≤ r + s ≤ ki+1 − 1−ψ−i 1k if r + s = −ki−ψ−i hi,−11k if r + s = −ki − 1, (U09)for any 1 ≤ i, j ≤ n− 1 and r, s such that the above relations make sense.Remark 2.2.2. Most of the relations in Definition 2.1.1 can not be substituted by q = 0 directly. From therelations (U3), (U4), in sln case we have cij = 2 when i = j. Taking q = 0, we can see that they become therelations (U04), (U05) when i = j. This is the main reason why we call it the name shifted q = 0 affine algebra.9Chapter 3Categorical actionIn this chapter, we give a definition of the categorical action for shifted q = 0 affine algebra that defined above.Again, we use the notation C(n,N) and αi defined in Chapter 2. We also denote by 〈·, ·〉 : Zn × Zn → Z thestandard pairing. The definition follows the similar formalism as in [CK2], which comes from the definition of(g, θ) action defined in [C].Definition 3.0.1. A categorical U˙0,N (Lsln) action consists of a target 2-category K, which is triangulated, C-linear and idempotent complete. The objects in K areOb(K) = {K(k) | k ∈ C(n,N)}where each K(k) is also a triangulated category, and each Hom space Hom(K(k),K(l)) is also triangulated.The morphisms are given by1. 1-morphisms: Ei,r1k = 1k+αiEi,r, Fi,s1k = 1k−αiFi,s, (Ψ±i )±11k = 1k(Ψ±i )±1, Hi,±11k = 1kHi,±1,where 1 ≤ i ≤ n− 1, −ki − 1 ≤ r ≤ 0, 0 ≤ s ≤ ki+1 + 1. Here 1k is the identity functor of K(k).Subject to the following relations.1. The space of maps between any two 1-morphisms is finite dimensional.2. If α = αi or α = αi + αj for some i, j with 〈αi, αj〉 = −1, then 1k+rα = 0 for r  0 or r  0.3. Suppose i 6= j. If 1k+αi and 1k+αj are nonzero, then 1k and 1k+αi+αj are also nonzero.4. Hi,±11k are adjoint to each other, i.e. (Hi,11k)L ∼= 1kHi,−1 ∼= (Hi,11k)R.105. Ei,r and Fi,r are left and right adjoints to each other up to twisted by the functors Ψ±i . More precisely,(a) (Ei,r1k)R ∼= 1kFi,−r+ki+1+1(Ψ+i )−1 for all 1 ≤ i ≤ n− 1(b) (Ei,r1k)L ∼= 1k(Ψ−i )−1Fi,−r−ki for all 1 ≤ i ≤ n− 1.6. (Ψ±i )±1(Ψ±j )±11k ∼= (Ψ±j )±1(Ψ±i )±11k for all i, j.7. Hi,±1Hj,±11k ∼= Hj,±1Hi,±11k for all i, j.8. Hi,±1Ψ±j 1k ∼= Ψ±j Hi,±11k for all i, j.9. The relations between Ei,r, Ej,s are given by the following(a) We haveEi,r+1Ei,s1k ∼=Ei,s+1Ei,r1k[−1] if r − s ≥ 10 if r = sEi,s+1Ei,r1k[1] if r − s ≤ −1.(b) Ei,r, Ei+1,s would related by the following exact triangle→ Ei+1,sEi,r+11k → Ei+1,s+1Ei,r1k → Ei,rEi+1,s+11k → .(c) We haveEi,rEj,s1k ∼= Ej,sEi,r1k, if |i− j| ≥ 2.10. The relations between Fi,r, Fj,s are given by the following(a) We haveFi,rFi,s+11k ∼=Fi,sFi,r+11k[1] if r − s ≥ 10 if r = sFi,sFi,r+11k[−1] if r − s ≤ −1.(b) Fi,r, Fi+1,s are related by the following exact triangles→ Fi,r+1Fi+1,s1k → Fi,rFi+1,s+11k → Fi+1,s+1Fi,r1k → .(c) We haveFi,rFj,s1k ∼= Fj,sFi,r1k, if |i− j| ≥ 2.1111. The relations between Ei,r, Ψ±j are given by the following(a) For i = j, we haveΨ±i Ei,r1k ∼= Ei,r+1Ψ±i 1k[∓1].(b) For |i− j| = 1, we have the followingΨ±i Ei±1,r1k ∼= Ei±1,r−1Ψ±i 1k[±1]Ψ±i Ei∓1,r1k ∼= Ei∓1,rΨ±i 1k.(c) For |i− j| ≥ 2, we haveΨ±i Ej,r1k ∼= Ej,rΨ±i 1k.12. The relations between Fi,r, Ψ±j are given by the following(a) For i = j, we haveΨ±i Fi,r1k ∼= Fi,r−1Ψ±i 1k[±1].(b) For |i− j| = 1, we have the followingΨ±i Fi±1,s1k ∼= Fi±1,s+1Ψ±i 1k[∓1]Ψ±i Fi∓1,r1k ∼= Fi∓1,rΨ±i 1k.(c) For |i− j| ≥ 2, we haveΨ±i,Fj,r1k ∼= Fj,rΨ±i 1k.13. The relations between Ei,r, Hj,±1 are given by the following(a) For i = j, they are related by the following exact triangles→ Hi,1Ei,r1k → Ei,rHi,11k → (Ei,r+1⊕Ei,r+1[1])1k →→ Ei,rHi,−11k → Hi,−1Ei,r1k → (Ei,r−1⊕Ei,r−1[1])1k → .(b) For |i− j| = 1, they are related by the following exact triangles→ Hi,1Ei+1,r1k → Ei+1,rHi,11k → Ei+1,r+11k →12→ Ei−1,r+11k → Hi,1Ei−1,r1k → Ei−1,rHi,11k →→ Ei+1,r−11k → Ei+1,rHi,−11k → Hi,−1Ei+1,r1k →→ Ei−1,rHi,−11k → Hi,−1Ei−1,r1k → Ei−1,r−11k → .(c) For |i− j| ≥ 2, we haveHi,±1Ej,r1k ∼= Ej,rHi,±11k.14. The relations between Fi,r, Hj,±1 are given by the following(a) For i = j, they are related by the following exact triangles→ Fi,rHi,11k → Hi,1Fi,r1k → (Fi,r+1⊕Fi,r+1[1])1k →→ Hi,−1Fi,r1k → Fi,rHi,−11k → (Fi,r−1⊕Fi,r−1[1])1k → .(b) For |i− j| = 1, they are related by the following exact triangles→ Fi+1,rHi,11k → Hi,1Fi+1,r1k → Fi+1,r+11k →→ Fi−1,r+11k → Fi−1,rHi,11k → Hi,1Fi−1,r1k →→ Fi+1,r−11k → Hi,−1Fi+1,r1k → Fi+1,rHi,−11k →→ Hi,−1Fi−1,r1k → Fi−1,rHi,−11k → Fi−1,r−11k → .(c) For |i− j| ≥ 2, we haveHi,±1Fj,r1k ∼= Fj,rHi,±11k.15. If i 6= j, then Ei,rFj,s1k ∼= Fj,sEi,r1k for all r, s ∈ Z.16. For Ei,rFi,s1k,Fi,sEi,r1k ∈ Hom(K(k),K(k)), they are related by exact triangles, more precisely,(a)→ Fi,sEi,r1k → Ei,rFi,s1k → Ψ+i Hi,11k → Fi,sEi,r1k[1]→ if r + s = ki+1 + 1(b)→ Fi,sEi,r1k → Ei,rFi,s1k → Ψ+i 1k → Fi,sEi,r1k[1]→ if r + s = ki+113(c)→ Ei,rFi,s1k → Fi,sEi,r1k → Ψ−i 1k → Ei,rFi,s1k[1]→ if r + s = −ki(d)→ Ei,rFi,s1k → Fi,sEi,r1k → Ψ−i Hi,−11k → Ei,rFi,s1k[1]→ if r + s = −ki − 1(e)Fi,sEi,r1k ∼= Ei,rFi,s1k if − ki + 1 ≤ r + s ≤ ki+1 − 1.We give some remarks about this definition.Remark 3.0.2. The 2-categoryK is called idempotent complete if for any 2-morphism f with f2 = f , the imageof f is contained in K.Remark 3.0.3. Note that in our definition of categorical action, we do not have the linear mapsSpan{αi|1 ≤ i ≤ n− 1} → End2(1k), k ∈ C(n,N)which is used to give the element θ in the definition of (g, θ) or (gˆ, θ) action in [C] or [CL]. This is becauseusually the geometry of the spaces that appear in our setting does not have (nontrivial) flat deformation, seeChapter 5 for examples. The data of flat deformation can be used to obtain linear map Span{αi|1 ≤ i ≤n− 1} → End2(1k), which was showed in [C].14Chapter 4Preliminaries on coherent sheaves andFourier-Mukai kernelsIn this chapter, we briefly recall some facts about Fourier-Mukai kernels and other tools that we would use forproofs in later chapters. The readers can consult the book by Huybrechts [H] for details.We will be working with the bounded derived category of coherent sheaves on X , which we denote it byDb(X). Throughout this article, all pullbacks, pushforwards, Homs, and tensor products of sheaves will bederived functors.LetX , Y be two smooth projective varieties. A Fourier-Mukai kernel is any object P in the derived categoryof coherent sheaves onX×Y . Given P ∈ Db(X×Y ), we may define the associated Fourier-Mukai transform,which is the functorΦP : Db(X)→ Db(Y )F 7→ pi2∗(pi∗1(F)⊗ P)We call ΦP the Fourier-Mukai transform with (Fourier-Mukai) kernel P . For convenience, we would just writeFM for Fourier-Mukai. The first property of FM transforms is that they have left and right adjoints that arethemselves FM transforms.Proposition 4.0.1. ([H] Proposition 5.9) For ΦP : Db(X)→ Db(Y ) is the FM transform with kernel P , definePL = P∨ ⊗ pi∗2ωY [dimY ], PR = P∨ ⊗ pi∗1ωX [dimX].ThenΦPL : Db(Y )→ Db(X), ΦPR : Db(Y )→ Db(X)15are the left and right adjoints of ΦP , respectively.The second property is the composition of FM transforms is also a FM transform.Proposition 4.0.2. ([H] Proposition 5.10) Let X,Y, Z be smooth projective varieties over C. Consider objectsP ∈ Db(X × Y ) and Q ∈ Db(Y × Z). They define FM transforms ΦP : Db(X) → Db(Y ), ΦQ : Db(Y ) →Db(Z). We would use ∗ to denote the operation for convolution, i.e.Q ∗ P := pi13∗(pi∗12(P)⊗ pi∗23(Q))Then forR = Q ∗ P ∈ Db(X × Z), we have ΦQ ◦ ΦP ∼= ΦR.Remark 4.0.3. Moreover by [H] remark 5.11, we have (Q∗P)L ∼= (P)L ∗ (Q)L and (Q∗P)R ∼= (P)R ∗ (Q)R.The next result we will use in later chapters is about tensor product of structure sheaves when their intersec-tion is transverse. Since we calculate many convolution of kernels, and from the definition we have to deal withmany tensor products of sheaves.Lemma 4.0.4. Let Z be a smooth projective variety, and X,Y ⊂ Z be smooth closed subvarieties, and i :X ↪→ Z, j : Y ↪→ Z r : X ∩ Y ↪→ Z be the closed embeddings. Consider OX ∈ Db(X), and OY ∈ Db(Y ), ifX and Y intersect transversally in Z, then we have i∗OX ⊗ j∗OY = r∗OX∩Y .The final result we will need in later is about derived pushforward of coherent sheaves. Let V be a vectorbundle of rank n on a variety X , where n ≥ 2. Then we can form the projective bundle P(V). We get in thisway a Pn−1 fibration pi : P(V) → X . Let OP(V)(−1) be the relative tautological bundle and OP(V)(1) be thedual bundle, we have OP(V)(i) = OP(V)(1)⊗i for i ∈ Z. Then we have the following result.Proposition 4.0.5.pi∗OP(V)(i) ∼=Symi(V∨) if i ≥ 00 if 1− n ≤ i ≤ −1Sym−i−n(V)⊗ det(V)[1− n] if i ≤ −n(4.0.1)in the derived category Db(X), where V∨ = RHom(V,OX) ∈ Db(X).Proof. Let DP(V) and DX be the Verdier dual functors for the derived categories Db(P(V)) and Db(X), respec-tively. Then since pi is proper, we get pi∗DP(V ) ∼= DXpi∗.16By definition, we have DP(V)OP(V) = O∨P(V) ⊗ ωP(V)[dimP(V)], where ωP(V) is the canonical bundle onP(V). Also, DXOX = O∨X ⊗ ωX [dimP(X)]. Since that pi∗(OP(V)) = OX , and note that O∨X ∼= OX , O∨P(V) ∼=OP(V), we getpi∗(O∨P(V) ⊗ ωP(V))[dimP(V)] ∼= O∨X ⊗ ωX [dimX]thuspi∗(ωP(V ))[dimP(V )] ∼= ωX [dimX]. (4.0.2)We have the short exact sequence0→ OP(V)(−1)→ OP(V) ⊗ pi∗(V)→ Q→ 0 (4.0.3)whereQ is the quotient bundle. The relative tangent bundle is TP(V)/X = HomP(V)(OP(V)(−1), Q) ∼= OP(V)(1)⊗Q. Thus we haveQ ∼= TP(V)/X⊗OP(V)(−1), and the relative cotangent bundle is ΩP(V)/X ∼= Q∨⊗OP(V)(−1).Tensoring the above short exact sequence (4.0.3) by OP(V)(1) and taking duals, we get0→ ΩP(V)/X → OP(V)(−1)⊗ pi∗(V∨)→ O∨P(V) → 0.Thus we get that the relative canonical bundle is ωP(V)/X ∼= det(ΩP(V)/X) ∼= det(OP(V)(−1)⊗ pi∗(V∨)) ∼=pi∗(det(V∨)) ⊗ OP(V)(−n), since rankV = n. For ωP(V)/X ∼= pi∗(ω∨X) ⊗ ωP(V), we get pi∗(ω∨X) ⊗ ωP(V) ∼=pi∗(det(V∨))⊗OP(V)(−n). So, ωP(V) ∼= pi∗(ωX ⊗ det(V∨))⊗OP(V)(−n).Hence, we get thatpi∗(ωP(V))[dimP(V)] ∼= pi∗(pi∗(ωX ⊗ det(V∨))⊗OP(V)(−n))[dimP(V)]∼= ωX ⊗ det(V∨)⊗ pi∗(OP(V)(−n))[dimP(V )].( projection formula)So, (4.0.2) becomesωX [dimX] ∼= ωX ⊗ det(V∨)⊗ pi∗(OP(V)(−n))[dimP(V)]for dimP(V) = dimX + n− 1, we conclude thatpi∗(OP(V)(−n)) ∼= det(V)[1− n].This verifies the case where i = −n. For 1 − n ≤ i ≤ −1, since there is no cohomology on the fibre, it iseasy to see that pi∗(OP(V )(i)) = 0 for 1− n ≤ i ≤ −1.17Now, for i = −n− l where l ≥ 0, we havepi∗DP(V)(OP(V)(l)) ∼= DXpi∗(OP(V)(l))using the same calculation as above, we get thatpi∗(OP(V)(−n− l)) ∼= det(V)⊗ pi∗(OP(V)(l))∨[1− n].This means that if we know pi∗(OP(V)(l)) for l ≥ 1, then we know all the cases. Since H0(Pn,OPn(i)) ∼=Symi((Cn+1)∨) for all i ≥ 0, we get that pi∗(OP(V)(l)) ∼= Syml(V∨) for l ≥ 0. Thus pi∗(OP(V )(−n − l)) =det(V)⊗ Syml(V)[1− n] for all l ≥ 0.18Chapter 5A geometric exampleIn this chapter, we give an geometric example that satisfy the above definition of categorical U˙0,N (Lsln) action.That means we have to define categories K(k), 1-morphisms Ei,r1k, Fi,s1k, Hi,±11k, and (Ψ±i )±11k, which arefunctors between those categories.For each k ∈ C(n,N), we define the partial flag varietyFlk(CN ) := {V• = (0 = V0 ⊂ V1 ⊂ ... ⊂ Vn = CN )|dimVi/Vi−1 = ki for all i}.We denote Y (k) = Flk(CN ) and Db(Y (k)) to be the bounded derived categories of coherent sheaves onY (k). Those would be the objectsK(k) of the triangulated 2-categoryK in Definition 3.0.1. On Y (k) we denoteVi to be the tautological bundle whose fibre over a point (0 = V0 ⊂ V1 ⊂ ... ⊂ Vn = CN ) is Vi.To define those 1-morphisms Ei,r1k, Fi,s1k, Hi,±11k, (Ψ±i )±11k, we use the language of FM transforms,that means we would define them by using FM kernels. So we have to introduce more geometries.5.1 Correspondence and related varieties.We define correspondences W 1i (k) ⊂ Y (k)× Y (k + αi) byW 1i (k) := {(V•, V ′•) ∈ Y (k)× Y (k + αi)|Vj = V ′j for j 6= i, and V ′i ⊂ Vi}.Then we have the natural line bundle Vi/V ′i on W 1i (k).19Next, we introduce new varietiesXi(k) := {(V ′′′• , V•, V ′′• , V ′•) ∈ Y (k + αi)× Y (k)× Y (k)× Y (k − αi)|V ′′′i ⊂ Vi ⊂ V ′i , V ′′′i ⊂ V ′′i ⊂ V ′i , V ′′′j = Vj = V ′′j = V ′j ∀ j 6= i}.On Xi(k), we have the divisor Di(k) ⊂ Xi(k) that defined byDi(k) := {(V ′′′• , V•, V ′′• , V ′•) ∈ Y (k + αi)× Y (k)× Y (k)× Y (k − αi)|V ′′′i ⊂ Vi = V ′′i ⊂ V ′i , V ′′′j = Vj = V ′′j = V ′j ∀ j 6= i}which is cut out by the natural section of the line bundlesHom(V ′′i /V ′′′i ,V ′i/Vi) orHom(Vi/V ′′′i ,V ′I/V ′′i ). Moreprecisely, we have OXi(k)(−Di(k)) ∼= V ′′i /V ′′′i ⊗ (V ′i/Vi)−1 ∼= Vi/V ′′′i ⊗ (V ′i/V ′′i )−1.We also have the following short exact sequences0→ V ′′i /V ′′′i → V ′i/Vi → ODi(k) ⊗ V ′i/Vi → 00→ Vi/V ′′′i → V ′i/V ′′i → ODi(k) ⊗ V ′i/V ′′i → 0.Let pi : Xi(k)→ Yi(k) be the projection by forgetting V ′′′• and V ′• . HereYi(k) = {(V•, V ′′• , ) ∈ Y (k)× Y (k) |dimVi ∩ V ′′i ≥i∑l=1kl − 1, Vj = V ′′j ∀ j 6= i}.Let ti : Yi(k) → Y (k) × Y (k) be the inclusion and ∆ : Y (k) → Y (k) × Y (k) the diagonal map, andι : W 1i (k) ↪→ Y (k) × Y (k + αi) be the inclusion. Then we define those 1-morphisms via using the abovegeometries.Definition 5.1.1. We define Ei,r1k, 1kFi,s, Hi,±11k, (Ψ±i )±11k to be FM transforms with the correspondingkernelsEi,r1k := ι∗(Vi/V ′i)r ∈ Db(Y (k)× Y (k + αi)),1kFi,r := ι∗(V ′i/Vi)r ∈ Db(Y (k + αi)× Y (k)),(Ψ+i )±11k := ∆∗det(Vi+1/Vi)±1[±(1− ki+1)] ∈ Db(Y (k)× Y (k)),(Ψ−i )±11k := ∆∗det(Vi/Vi−1)∓1[±(1− ki)] ∈ Db(Y (k)× Y (k)),Hi,11k := (Ψ+i 1k)−1 ∗ [ti∗pi∗(O2Di ⊗ (V ′i/Vi)ki+1+1)] ∈ Db(Y (k)× Y (k)),Hi,−11k := (Ψ−i 1k)−1 ∗ [ti∗pi∗(O2Di ⊗ (Vi/V ′′′i )−ki−1)] ∈ Db(Y (k)× Y (k)).20Then we have the following theorem.Theorem 5.1.2. Let K be the triangulated 2-categories whose nonzero objects are K(k) = Db(Y (k)) wherek ∈ C(n,N), the 1-morphisms are kernels defined in Definition 5.1.1 and the 2-morphisms are maps betweenkernels. Then this gives a categorical U˙0,N (Lsln) action.We devote the rest of this chapter to a proof of this theorem.5.2 sl2 caseSince many relations in the Definition 3.0.1 of categorical U˙0,N (Lsln) action can be reduced to the sl2 case, wewill prove there is a categorical U˙0,N (Lsl2) action first.Now n = 2 and 1 ≤ i ≤ n − 1 imply we only have i = 1. To simplify the notation, we would drop i fromall the functors with i in their notation. Also, we have k = (k1, k2) with k1 + k2 = N , the partial flag varietyFlk(CN ) are just the GrassmanniansG(k,N) = {0 ⊂ V ⊂ CN | dimV = k},and the correspondences W 11 ((k,N − k)) are the 2-steps partial flag varietiesFl(k − 1, k) = {0 ⊂ V ′ ⊂ V ⊂ CN | dimV ′ = k − 1, dimV = k}.Here we would just write (k,N − k) with 0 ≤ k ≤ N for k. Thus we have the 1-morphisms Er1(k,N−k),Fs1(k,N−k), H±11(k,N−k), (Ψ±)±11(k,N−k).We show thatTheorem 5.2.1. The data above define a categorical U˙0,N (Lsl2) action.To prove this, we need to prove the relations (1), (4), (5) (9a), (10a), (11a), (12a), (13a), (14a) and (16) inthe Definition 3.0.1.Note that relation (1) is obvious. So we check the rest. Before we check them, let us remark that sinceall functors above are defined by using kernels, by Proposition 4.0.2, composition of functors corresponds toconvolution of kernels, and therefore we may check the relations in terms of kernels.Many of the relations that we prove below are for Er1(k,N−k), and most of the proof involve the calculation21of convolution of kernels. So it is helpful to have the following picture.G(k,N)×G(k − 1, N)pi1||pi2##Fl(k − 1, k)p1uup2))ιOOG(k,N) G(k − 1, N)where ι : Fl(k − 1, k) → G(k,N) × G(k − 1, N) is the natural inclusion and pi1, pi2, p1, p2 are the naturalprojections.The first is relation (5).Lemma 5.2.2.(Er1(k,N−k))R ∼= 1(k,N−k)F−r+N−k+1 ∗ (Ψ+)−1(Er1(k,N−k))L ∼= 1(k,N−k)(Ψ−)−1 ∗ F−r−k.Proof. It suffices to prove the first case, the other is similar.By Proposition 4.0.1, we have the right adjoint of Er1(k,N−k) is given by{ι∗(V/V ′)r}∨ ⊗ pi∗1(ωG(k,N))[dimG(k,N)].Thus it suffices to calculate {ι∗(V/V ′)r}∨. For Fl(k−1, k) ⊂ G(k,N)×G(k−1, N) the derived dual is givenby{ι∗(V/V ′)r}∨ ∼= ι∗((V/V ′)r ⊗ ωFl(k−1,k))⊗ ω−1G(k,N)×G(k−1,N)[−codimFl(k − 1, k)].We have ω−1G(k,N)×G(k−1,N)∼= pi∗1(ω−1G(k,N)) ⊗ pi∗2(ω−1G(k−1,N)). To calculate ωFl(k−1,k), considering theprojection pi : Fl(k − 1, k) → G(k − 1, N). We have ωFl(k−1,k) ∼= ωrel ⊗ pi∗ωG(k−1,N). So summarizing wehave{ι∗(V/V ′)r}∨ ⊗ pi∗1(ωG(k,N))[dimG(k,N)]∼= ι∗(ωrel ⊗ (V/V ′)−r)[dimG(k,N)− codimFl(k − 1, k)] (projection formula).22For ωrel, there is a short exact sequence of vector bundles on Fl(k − 1, k)0→ V/V ′ → CN/V ′ → CN/V → 0this gives the relative cotangent bundleT∨rel ∼= V/V ′ ⊗ (CN/V)∨ andωrel ∼=N−k∧V/V ′ ⊗ (CN/V)∨ ∼= (V/V ′)N−k ⊗ det(CN/V)−1 ∼= (V/V ′)N−k+1 ⊗ det(CN/V ′)−1.A calculation gives dimG(k,N)− codimFl(k − 1, k) = N − k. Thus{ι∗(V/V ′)r}∨ ⊗ pi∗1(ωG(k,N))[dimG(k,N)] ∼= ι∗((V/V ′)−r+N−k+1 ⊗ det(CN/V ′)−1)[N − k].Note that Ψ+1(k−1,N−k+1) is defined by tensoring det(CN/V ′) and then shifted by [−N + k]. So it is thekernel 1(k,N−k)F−r+N−k+1 ∗ (Ψ+)−1.Next, we verify relations (11a), (12a).Lemma 5.2.3.(Ψ± ∗ Er)1(k,N−k) ∼= (Er+1 ∗Ψ±)1(k,N−k)[∓1](Ψ± ∗ Fr)1(k,N−k) ∼= (Fr−1 ∗Ψ±)1(k,N−k)[±1].Proof. It suffices to prove the first case, the others are similar. We have(Ψ+ ∗ Er)1(k,N−k) ∼= pi13∗(pi∗12Er1(k,N−k) ⊗ pi∗23Ψ+1(k−1,N−k+1)) (5.2.1)∼= pi13∗(pi∗12ι∗(V/V ′)r ⊗ pi∗23∆∗det(CN/V ′)[k −N ]) (5.2.2)and the following commutative diagramsFl(k − 1, k)×G(k − 1, N) G(k,N)×G(k − 1, N)×G(k − 1, N)Fl(k − 1, k) G(k,N)×G(k − 1, N)ι×idq1 pi12ι23G(k,N)×G(k − 1, N) G(k,N)×G(k − 1, N)×G(k − 1, N)G(k − 1, N) G(k − 1, N)×G(k − 1, N)id×∆pi2 pi23∆where q1 is the natural projection. Using base change, (5.2.2) becomespi13∗((ι× id)∗q∗1(V/V ′)r ⊗ (id×∆)∗pi∗2det(CN/V ′)[k −N ]) (5.2.3)∼= pi13∗(id×∆)∗((id×∆)∗(ι× id)∗q∗1(V/V ′)r ⊗ pi∗2det(CN/V ′)[k −N ]) (5.2.4)∼= (id×∆)∗(ι× id)∗q∗1(V/V ′)r ⊗ pi∗2det(CN/V ′)[k −N ] (5.2.5)Note that here we use pi13 ◦ (id×∆) = id. Next, we have the following commutative diagramFl(k − 1, k) G(k,N)×G(k − 1, N)Fl(k − 1, k)×G(k − 1, N) G(k,N)×G(k − 1, N)×G(k − 1, N)ιid×p2 id×∆ι×idSo (5.2.5) becomesι∗(id× p2)∗q∗1(V/V ′)r ⊗ pi∗2det(CN/V ′)[k −N ] (5.2.6)∼= ι∗(V/V ′)r ⊗ pi∗2det(CN/V ′)[k −N ] (5.2.7)Again, here we use q1 ◦ (id× p2) = id. Thus (Er+1 ∗Ψ+)1(k,N−k) is given byι∗(V/V ′)r+1 ⊗ pi∗1 det(CN/V)[1 + k −N ] ∼= ι∗((V/V ′)r+1 ⊗ det(CN/V))[1 + k −N ].Similarly, (Ψ+ ∗ Er)1(k,N−k) is given bypi∗2 det(CN/V ′)⊗ ι∗(V/V ′)r[k −N ] ∼= ι∗(det(CN/V ′)⊗ (V/V ′)r)[k −N ].Note that on Fl(k − 1, k), we have det(CN/V ′) ∼= V/V ′ ⊗ det(CN/V). Combining these, the resultfollows.Next, we verify relations (9a), (10a).24Lemma 5.2.4.(Er+1 ∗ Es)1(k,N−k) ∼=(Es+1 ∗ Er)1(k,N−k)[−1] if r − s ≥ 10 if r = s(Es+1 ∗ Er)1(k,N−k)[1] if r − s ≤ −1(Fr ∗ Fs+1)1(k,N−k) ∼=(Fs ∗ Fr+1)1(k,N−k)[1] if r − s ≥ 10 if r = s(Fs ∗ Fr+1)1(k,N−k)[−1] if r − s ≤ −1.Proof. It suffices to prove the first case, the others are similar.For the first one, we have r − s ≥ 1. Then (Er+1 ∗ Es)1(k,N−k) is given byι′∗(V ′/V ′′)r+1 ∗ ι∗(V/V ′)s = pi13∗(pi∗12ι∗(V/V ′)s ⊗ pi∗23ι′∗(V ′/V ′′)r+1) (5.2.8)Here ι′ : Fl(k − 2, k − 1) → G(k − 1, N) × G(k − 2, N) is the natural inclusion. We have the followingcommutative diagramsFl(k − 1, k)×G(k − 2, N) G(k,N)×G(k − 1, N)×G(k − 2, N)Fl(k − 1, k) G(k,N)×G(k − 1, N)ι×idr1 pi12ιG(k,N)× Fl(k − 2, k − 1) G(k,N)×G(k − 1, N)×G(k − 2, N)Fl(k − 2, k − 1) G(k − 1, N)×G(k − 2, N)id×ι′r2 pi23ι′where r1, r2 are the natural projections. So (5.2.8) becomespi13∗((ι× id)∗r∗1(V/V ′)s ⊗ (id× ι′)∗r∗2(V ′/V ′′)r+1) (5.2.9)∼= pi13∗(ι× id)∗(r∗1(V/V ′)s ⊗ (ι× id)∗(id× ι′)∗r∗2(V ′/V ′′)r+1). (5.2.10)25We have the following commutative diagramFl(k − 2, k − 1, k) G(k,N)× Fl(k − 2, k − 1)Fl(k − 1, k)×G(k − 2, N) G(k,N)×G(k − 1, N)×G(k − 2, N)t2t1 id×ι′ι×idwhere t1, t2 are the natural maps. So (5.2.10) becomespi13∗(ι× id)∗(r∗1(V/V ′)s ⊗ t1∗t∗2r∗2(V ′/V ′′)r+1) (5.2.11)∼= pi13∗(ι× id)∗t1∗(t∗r∗1(V/V ′)s ⊗ t∗2r∗2(V ′/V ′′)r+1) (5.2.12)∼= pi13∗(ι× id)∗t1∗((V ′/V ′′)r+1−s ⊗ det(V/V ′′)s) (5.2.13)Finally we have the following commutative diagramFl(k − 2, k − 1, k) G(k,N)×G(k − 1, N)×G(k − 2, N)Fl(k − 2, k) G(k,N)×G(k − 2, N)(ι×id)◦t1pi pi13ι′′Here ι′′ is the natural inclusion and pi : Fl(k − 2, k − 1, k) → Fl(k − 2, k) is a P1-fibration. We haveV ′/V ′′ ∼= OP(V/V”)(−1). Since r − s ≥ 1, we get −r + s − 1 ≤ −2 and using Proposition 4.0.5, we get that(5.2.13) becomesι′′∗pi∗(OP(V/V ′′)(−r + s− 1)⊗ det(V/V ′′)s)∼= ι′′∗(Symr−s−1(V/V ′′)⊗ det(V/V ′′)s+1)[−1].Thus (Er+1 ∗ Es)1k is isomorphic to ι′′∗(Symr−s−1(V/V ′′)⊗ det(V/V ′′)s+1)[−1].Similar calculation tells us that (Es+1 ∗Er)1(k,N−k) is given by ι′′∗(Symr−s−1(V/V ′′)⊗det(V/V ′′)s+1) andthe relation follows.Next, we prove the relation (16), before we move to the proof, we would like to use some of the aboveproperties to reduce the cases to simpler ones.Note that from Lemma 5.2.3, we have (Ψ+ ∗ Er)1(k,N−k) ∼= (Er+1 ∗Ψ+)1(k,N−k)[−1]. Since Ψ+1(k,N−k)26is invertible, we get[Ψ+ ∗ Er ∗ (Ψ+)−1]1(k,N−k) ∼= Er+11(k,N−k)[−1].Since it is true for all r ∈ Z, we can apply this inductively, and we get[(Ψ+)r ∗ E ∗ (Ψ+)−r]1(k,N−k) ∼= Er1(k,N−k)[−r]where (Ψ+)r means (Ψ+) convolution with itself r times.Similarly, for Fs1(k,N−k), we have[(Ψ+)−s ∗ F ∗ (Ψ+)s]1(k,N−k) ∼= Fs1(k,N−k)[−s].Then we have(Er ∗ Fs)1(k,N−k) ∼= [(Ψ+)r ∗ E ∗ (Ψ+)−r ∗ Fs]1(k,N−k)[r] ∼= [(Ψ+)r ∗ E ∗ Fs+r ∗ (Ψ+)−r]1(k,N−k).On the other hand(Fs ∗ Er)1(k,N−k) ∼= [Fs ∗ (Ψ+)r ∗ E ∗ (Ψ+)−r]1(k,N−k)[r] ∼= [(Ψ+)r ∗ Fr+s ∗ E ∗ (Ψ+)−r]1(k,N−k).From this, since Ψ+1(k,N−k) is invertible, in order to compare (Fs ∗ Er)1(k,N−k) and (Er ∗ Fs)1(k,N−k), itsuffices to compare (Fr+s ∗ E)1(k,N−k) and (E ∗ Fr+s)1(k,N−k).Thus, it suffices prove the following proposition.Proposition 5.2.5. We have the following exact triangles in Db(G(k,N)×G(k,N)).→ (Fr+s ∗ E)1(k,N−k) → (E ∗ Fr+s)1(k,N−k) → (Ψ+ ∗ H1)1(k,N−k) → (Fr+s ∗ E)1(k,N−k)[1]→, r + s = N − k + 1→ (Fr+s ∗ E)1(k,N−k) → (E ∗ Fr+s)1(k,N−k) → Ψ+1(k,N−k) → (Fr+s ∗ E)1(k,N−k)[1]→, r + s = N − k→ (E ∗ Fr+s)1(k,N−k) → (Fr+s ∗ E)1(k,N−k) → Ψ−1(k,N−k) → (E ∗ Fr+s)1(k,N−k)[1]→, r + s = −k→ (E ∗ Fr+s)1(k,N−k) → (Fr+s ∗ E)1(k,N−k) → (Ψ− ∗ H−1)1(k,N−k) → (E ∗ Fr+s)1(k,N−k)[1]→, r + s = −k − 1(Fr+s ∗ E)1(k,N−k) ∼= (E ∗ Fr+s)1(k,N−k), −k + 1 ≤ r + s ≤ N − k − 1.The first thing we would deal with is when r + s = 0, which is the following lemma.Lemma 5.2.6. (E ∗ F)1(k,N−k) ∼= (F ∗ E)1(k,N−k)27Proof. We have to compute the convolution of kernels. Starting from (E ∗ F)1(k,N−k), the kernel is given bypi13∗(pi∗12τ∗OFl(k,k+1) ⊗ pi∗23ι∗OFl(k,k+1)) (5.2.14)where τ : Fl(k, k+ 1)→ G(k,N)×G(k+ 1, N) and ι : Fl(k, k+ 1)→ G(k+ 1)×G(k,N) are the naturalinclusions. Also piij are natural projections from G(k,N) × G(k + 1, N) × G(k,N) to the corresponding(i, j)-components.We have the following commutative diagramFl(k, k + 1)×G(k,N) G(k,N)×G(k + 1, N)×G(k,N)Fl(k, k + 1) G(k,N)×G(k + 1, N)τ×ida1 pi12τG(k,N)× Fl(k, k + 1) G(k,N)×G(k + 1, N)×G(k,N)Fl(k, k + 1) G(k + 1, N)×G(k,N)id×ιa2 pi23ιwhere a1, a2 are the natural projections.Using base change, (5.2.14) becomespi13∗((τ × id)∗a∗1OFl(k,k+1) ⊗ (id× ι)∗a∗2OFl(k,k+1)) (5.2.15)∼= pi13∗((τ × id)∗OFl(k,k+1)×G(k,N) ⊗ (id× ι)∗OG(k,N)×Fl(k,k+1)) (5.2.16)We have the following commutative diagramZ Fl(k, k + 1)×G(k,N)G(k,N)× Fl(k, k + 1) G(k,N)×G(k + 1, N)×G(k,N)b1b2 τ×idid×ιwhere Z is the intersectionZ := (Fl(k, k + 1)×G(k,N)) ∩ (G(k,N)× Fl(k, k + 1))= {(V, V ′, V ′′)| dimV = dimV ′′ = k, dimV ′ = k + 1, and V, V ′′ ⊂ V ′}.28Then (5.2.16) becomespi13∗(id× ι)∗((id× ι)∗(τ × id)∗OFl(k,k+1)×G(k,N) ⊗OG(k,N)×Fl(k,k+1))∼= pi13∗(id× ι)∗(b2∗b∗1OFl(k,k+1)×G(k,N) ⊗OG(k,N)×Fl(k,k+1))∼= pi13∗(id× ι)∗b2∗(b∗1OFl(k,k+1)×G(k,N) ⊗ b∗2OG(k,N)×Fl(k,k+1))∼= pi13∗(id× ι)∗b2∗(OZ ⊗OZ) ∼= pi13∗(id× ι)∗b2∗(OZ)Finally, we have the following commutative diagramZ G(k,N)×G(k + 1, N)×G(k,N)Y G(k,N)×G(k,N)j1pi13|Z pi13twhere Y = pi13(Z) = {(V, V ′′) | dimV ∩V ′′ ≥ k− 1} is the image of Z under pi13 and j1, t are the inclusions.For pi13 : G(k,N) × G(k + 1, N) × G(k,N) → G(k,N) × G(k,N), we have when pi13 restrict to Z itbecomes pi13|Z : Z → G(k,N)×G(k,N) with fibres over (V, V ′′) ∈ G(k,N)×G(k,N) equal to(pi13|Z)−1(V, V ′′) =PN−k−1 if V = V ′′point if dimV ∩ V ′′ = k − 1φ if dimV ∩ V ′′ ≤ k − 2.Here Y ⊂ G(k,N)×G(k,N) is a Schubert variety, so it has rational singularities. This gives that.pi13∗(id× ι)∗b2∗(OZ) ∼= t∗pi13∗(OZ) ∼= t∗OYThus (E ∗ F)1(k,N−k) ∼= t∗OY .Using the same method for calculating (F ∗ E)1(k,N−k), we would end up with the following commutativediagramZ ′ G(k,N)×G(k − 1, N)×G(k,N)Y G(k,N)×G(k,N)j2pi13′ |Z′ pi13′t29where Z ′ is the intersectionZ ′ := (Fl(k − 1, k)×G(k,N)) ∩ (G(k,N)× Fl(k − 1, k))= {(V, V ′′′, V ′′)| dimV = dimV ′′ = k,dimV ′′′ = k − 1, and V ′′′ ⊂ V, V ′′}.and j2 is the inclusion.Restricting pi13′ to Z ′ we get pi13′ |Z′ : Z ′ → G(k,N) × G(k,N). The fibre over (V, V ′′) ∈ G(k,N) ×G(k,N) is(pi13′ |Z′)−1(V, V ′′) =Pk−1 if V = V ′′point if dimV ∩ V ′′ = k − 1φ if dimV ∩ V ′′ ≤ k − 2.Again, we have pi13(Z ′) = {(V, V ′′)|dimV ∩ V ′′ ≥ k − 1} = Y , which is the same as we got in the(E ∗ F)1(k,N−k) case and it is a Schubert variety. Thus pi13′∗(OZ′) = OY and (F ∗ E)1(k,N−k) ∼= t∗OY .Now we move to the case where r+ s is nonzero. To compare (E ∗Fr+s)1k and (Fr+s ∗ E)1k, by using theabove commutative diagrams, we have to compare the following two objectspi13∗(j1∗(V ′/V)r+s) ∼= t∗pi13∗(V ′/V)r+s and pi13′∗(j2∗(V ′′/V ′′′)r+s) ∼= t∗pi13∗(V ′′/V ′′′)r+sin the derived category Db(G(k,N)×G(k,N)).Note that both are pushforwards to Y . In order to handle the case where we tensor non-trivial line bundles,instead of directly pushing forward to Y , we may lift the line bundles to a much larger space, e.g., their fibredproduct. The fibred product space X := Z ×Y Z ′, is given byX = {(V ′′′, V, V ′′, V ′)|dim(V ∩ V ′′) ≥ k − 1, V ′′′ ⊂ V, V ′′ ⊂ V ′}.We have the following diagramX = Z ×Y Z ′ ZZ ′ Yp1p2 pi13pi13′(5.2.17)where p1 and p2 are defined byp1((V′′′, V, V ′′, V ′)) = (V, V ′, V ′′), p2((V ′′′, V, V ′′, V ′)) = (V, V ′′′, V ′′).30Moreover, we also have the natural projection p : X → Y that is defined by p((V ′′′, V, V ′′, V ′)) = (V, V ′′).On X , we denote by D ⊂ X to be the locus where V = V ′′, it is easy to see thatD = Fl(k − 1, k, k + 1) = {(V ′′′, V, V ′)|V ′′′ ⊂ V ⊂ V ′}.Recall that we had used the space X to defined the Fourier-Mukai transform H±11(k,N−k). So the sameas before, we have OX(−D) ∼= V ′′/V ′′′ ⊗ (V ′/V)−1 ∼= V/V ′′′ ⊗ (V ′/V ′′)−1, and the following short exactsequences0→ V ′′/V ′′′ → V ′/V → OD ⊗ V ′/V → 00→ V/V ′′′ → V ′/V ′′ → OD ⊗ V ′/V ′′ → 0.Furthermore, we can relate this to Proposition 4.0.5. Clearly, we have p(D) = ∆, and note that the diagonal∆ ⊂ Y is just the Grassmannians, i.e. ∆ = G(k,N). The preimage of the diagonal ∆ under pi13 ispi−113 (∆) = {(V, V ′, V ′′)|V = V ′′, V ⊂ V ′,dimV ′ = k + 1} = {(V, V ′) : V ⊂ V ′} = Fl(k, k + 1).There is a tautological quotient bundle on G(k,N), i.e. CN/V , where V is the tautological bundle of rank k onG(k,N). Taking the projective bundle P(CN/V), we obtain a PN−k−1-fibration P(CN/V)→ G(k,N) and weget P(CN/V) = Fl(k, k + 1).On P(CN/V), we have the relative tautological bundle OP(CN/V)(−1), and the associated relative Eulersequence0→ OP(CN/V)(−1)→ CN/V → Trel ⊗OP(CN/V)(−1)→ 0where Trel is the relative tangent bundle. On Fl(k, k + 1), we have the relative Euler sequence0→ V ′/V → CN/V → CN/V ′ → 0and the relative tangent bundle is Trel ∼= Hom(V ′/V,CN/V ′) ∼= (V ′/V)∨ ⊗ CN/V ′. Thus we concludethat V ′/V = OP(CN/V)(−1). More precisely, the restriction of the line bundle V ′/V on Z to pi−113 (∆) =Fl(k, k + 1) = P(CN/V) isV ′/V|pi−113 (∆)=Fl(k,k+1) = OP(CN/V)(−1).Similarly, we have the preimage of the diagonal under pi13′ ispi−113′(∆) = {(V, V ′′′, V ′′)|V = V ′′, V ′′′ ⊂ V,dimV ′′′ = k − 1} = {(V ′′′, V ) : V ′′′ ⊂ V } = Fl(k − 1, k).31There is a tautological bundle on G(k,N), i.e. V∨. Taking the projective bundle P(V∨), we obtain a Pk−1-fibration P(V∨)→ G(k,N) and we get P(V∨) = Fl(k − 1, k).On P(V∨), we have the relative tautological bundle OP(V∨)(−1), and the associated relative Euler sequence0→ OP(V∨)(−1)→ V∨ → Trel ⊗OP(V∨)(−1)→ 0where Trel is the relative tangent bundle. On Fl(k − 1, k), we have the relative Euler sequence0→ V ′′′ → V → V/V ′′′ → 0taking dual, we get0→ (V/V ′′′)∨ → V∨ → V ′′′∨ → 0.Thus we conclude that (V/V ′′′)∨ = OP(V∨)(−1). More precisely, the restriction of the bundle (V/V ′′′)∨ on Z ′to pi′−113 (∆) = Fl(k − 1, k) = P(V∨) is(V/V ′′′)∨|pi′−113 (∆)=Fl(k−1,k) = OP(V∨)(−1).Now, go to X , by definition D is just the partial flag variety D = Fl(k − 1, k, k + 1) and we also haveD = p−11 (pi−113 (∆)) = p−12 (pi−113′(∆)). So on D, we get that OD ⊗ V ′/V ∼= OD ⊗ OP(CN/V)(−1) and OD ⊗(V/V ′′′)∨ ∼= OD ⊗OP(V∨)(−1) and the same for V ′/V ′′ and V ′′/V ′′′.Thus the above diagram (5.2.17) becomesD = Fl(k − 1, k, k + 1) ⊂ X Z ⊃ Fl(k, k + 1)Fl(k − 1, k) ⊂ Z ′ Y ⊃ ∆ = G(k,N)p1p2 pi13pi13′(5.2.18)This tells us that actually the above two short exact sequences are0→ V ′′/V ′′′ → V ′/V → OD ⊗OP(CN/V)(−1)→ 0 (5.2.19)0→ V/V ′′′ → V ′/V ′′ → OD ⊗OP(CN/V)(−1)→ 0. (5.2.20)Tensoring (5.2.19) and (5.2.20) by (V ′′/V ′′′)−1 ⊗ (V ′/V)−1 and (V/V ′′′)−1 ⊗ (V ′/V ′′)−1 respectively, weget0→ (V ′/V)−1 → (V ′′/V ′′′)−1 → OD ⊗OP(V∨)(−1)→ 0 (5.2.21)320→ (V ′/V ′′)−1 → (V/V ′′′)−1 → OD ⊗OP(V∨)(−1)→ 0. (5.2.22)Proof. Now we prove Proposition 5.2.5.We already settled the case r+s = 0. For (k,N −k), we only prove the three cases 1 ≤ r+s ≤ N −k−1,r + s = N − k and r + s = N − k + 1. The rest are similar.The first case is 1 ≤ r + s ≤ N − k − 1. Then the kernel (E ∗ Fr+s)1(k,N−k) is t∗pi13∗(V ′/V)r+s.Using the fibred product Diagram (5.2.18), we havet∗pi13∗(V ′/V)r+s ∼= t∗pi13∗p1∗p∗1(V ′/V)r+s ∼= t∗p∗(V ′/V)r+s.Similarly, the kernel (Fr+s ∗ E)1(k,N−k) becomest∗pi13′∗(V ′′/V ′′′)r+s ∼= t∗pi13′∗p2∗p∗2(V ′′/V ′′′)r+s ∼= t∗p∗(V ′′/V ′′′)r+sWe have to compare p∗(V ′/V)r+s and p∗(V ′′/V ′′′)r+s.Using the exact sequence (5.2.19) on X , tensoring it by (V ′/V)r+s−1, we get0→ V ′′/V ′′′ ⊗ (V ′/V)r+s−1 → (V ′/V)r+s → OD ⊗OP(CN/V)(−r − s)→ 0.Now applying the derived pushforward p∗, since 1 ≤ r+ s ≤ N − k− 1, we get p∗(OD ⊗OP(CN/V)(−r−s)) = 0 by Proposition 4.0.5. Thus we get...→ p∗[V ′′/V ′′′ ⊗ (V ′/V)r+s−1]→ p∗((V ′/V)r+s)→ 0→ ...This implies that (E ∗ Fr+s)1(k,N−k) ∼= i∗p∗[V ′′/V ′′′ ⊗ (V ′/V)r+s−1]. Next, tensoring it by (V ′′/V ′′′) ⊗(V ′/V)r+s−2, then we get0→ (V ′′/V ′′′)2⊗ (V ′/V)r+s−2 → V ′′/V ′′′⊗ (V ′/V)r+s−1 → OD⊗OP(CN/V)(−r− s+ 1)⊗OP(V∨)(1)→ 0.Applying the derived pushforward p∗, to calculate p∗(OD⊗OP(CN/V)(−r− s+ 1)⊗OP(V∨)(1)), we using33the projection formula.p∗(OD ⊗OP(CN/V)(−r − s+ 1)⊗OP(V∨)(1)) ∼= pi13∗p1∗(OD ⊗ p∗1(OP(CN/V)(−r − s+ 1))⊗OP(V∨)(1))∼= pi13∗(OFl(k,k+1) ⊗OP(CN/V)(−r − s+ 1)⊗ V)∼= pi13∗(OFl(k,k+1) ⊗OP(CN/V)(−r − s+ 1)⊗ pi∗13(V))∼= V ⊗ pi13∗(OP(CN/V)(−r − s+ 1)) ∼= 0where the last isomorphism is by Proposition 4.0.5 and 1 ≤ r + s ≤ N − k − 1.Thus we get...→ p∗[(V ′′/V ′′′)2 ⊗ (V ′/V)r+s−2]→ p∗(V ′′/V ′′′ ⊗ (V ′/V)r+s−1)→ 0→ ...Doing this inductively, we would get that at the last stage, tensoring it by (V ′′/V ′′′)r+s−1, we get0→ (V ′′/V ′′′)r+s → (V ′′/V ′′′)r+s−1 ⊗ V ′/V → OD ⊗OP(CN/V)(−1)⊗OP(V∨)(r + s− 1)→ 0.Applying pushforward p∗, using the similar argument we get p∗(V ′′/V ′′′)r+s ∼= p∗[(V ′′/V ′′′)r+s−1⊗V ′/V].So we get(E∗Fr+s)1(k,N−k) ∼= t∗p∗[V ′′/V ′′′⊗(V ′/V)r+s−1] ∼= t∗p∗[(V ′′/V ′′′)2⊗(V ′/V)r+s−2] ∼= ... ∼= (Fr+s∗E)1(k,N−k)which prove this case.Next, we consider the case where r + s = N − k. Again, we use the above short exact sequences (5.2.19)to calculate. From0→ V ′′/V ′′′ ⊗ (V ′/V)r+s−1 → (V ′/V)r+s → OD ⊗OP(CN/V)(−r − s)→ 0since −r − s = −N + k, applying p∗, by Proposition 4.0.5, we get the exact triangle...→ p∗[V ′′/V ′′′ ⊗ (V ′/V)r+s−1]→ p∗((V ′/V)r+s)→ j∗ det(CN/V)[1 + k −N ]→ ...where j : ∆ = G(k,N)→ Y is the inclusion, note that ∆ = t ◦ j.Next, using0→ (V ′′/V ′′′)2⊗ (V ′/V)r+s−2 → V ′′/V ′′′⊗ (V ′/V)r+s−1 → OD⊗OP(CN/V)(−r− s+ 1)⊗OP(V∨)(1)→ 0.34Since r + s = N − k, we have −r − s+ 1 = −N + k + 1, by projection formula, we show thatp∗(OD ⊗OP(CN/V)(−r − s+ 1)⊗OP(V∨)(1)) = 0.So we obtain the exact triangle...→ p∗[(V ′′/V ′′′)2 ⊗ (V ′/V)r+s−2]→ p∗[V ′′/V ′′′ ⊗ (V ′/V)r+s−1]→ 0→ ...This implies that p∗[(V ′′/V ′′′)2 ⊗ (V ′/V)r+s−2] ∼= p∗[V ′′/V ′′′ ⊗ (V ′/V)r+s−1]. Then the rest are the sameas before, doing this inductively we end up with the exact triangle0→ (V ′′/V ′′′)r+s → (V ′′/V ′′′)r+s−1 ⊗ V ′/V → OD ⊗OP(CN/V)(−1)⊗OP(V∨)(r + s− 1)→ 0.Applying the pushforward p∗, we get...→ p∗[(V ′′/V ′′′)r+s]→ p∗[(V ′′/V ′′′)r+s−1 ⊗ V ′/V]→ 0→ ...Thus we havet∗p∗[V ′′/V ′′′ ⊗ (V ′/V)r+s−1] ∼= t∗p∗[(V ′′/V ′′′)2 ⊗ (V ′/V)r+s−2] ∼= ... ∼= (Fr+s ∗ E)1(k,N−k)and the exact triangle...→ (Fr+s ∗ E)1(k,N−k) → (E ∗ Fr+s)1(k,N−k) → ∆∗ det(CN/V)[1 + k −N ]→ ...Since ∆∗ det(CN/V)[1 + k −N ] is Ψ+1k, we prove the second case.Finally, we are in the third case r + s = N − k + 1. Again, using the following short exact sequence0→ V ′′/V ′′′ ⊗ (V ′/V)r+s−1 → (V ′/V)r+s → OD ⊗OP(CN/V)(−r − s)→ 0.Now −r − s = −N + k − 1, applying p∗, by Proposition 4.0.5, we get the exact triangle...→ p∗[V ′′/V ′′′ ⊗ (V ′/V)r+s−1]→ p∗((V ′/V)r+s)→ j∗(CN/V ⊗ det(CN/V))[1 + k −N ]→ ...35Next, using0→ (V ′′/V ′′′)2⊗ (V ′/V)r+s−2 → V ′′/V ′′′⊗ (V ′/V)r+s−1 → OD ⊗OP(CN/V)(−r− s+ 1)⊗OP(V∨)(1)→ 0since −r − s+ 1 = −N + k, by the projection formula, we havep∗(OD ⊗OP(CN/V)(−r − s+ 1)⊗OP(V∨)(1)) = j∗(V ⊗ det(CN/V))[1 + k −N ].So we get the following exact triangle→ p∗[(V ′′/V ′′′)2 ⊗ (V ′/V)r+s−2]→ p∗[V ′′/V ′′′ ⊗ (V ′/V)r+s−1]→ j∗(V ⊗ det(CN/V))[1 + k −N ]→ ...Then we have the third short exact sequence by tensoring (5.2.19) with (V ′′/V ′′′)2 ⊗ (V ′/V)r+s−3,0→ (V ′′/V ′′′)3⊗(V ′/V)r+s−3 → (V ′′/V ′′′)2⊗(V ′/V)r+s−2 → OD⊗OP(CN/V)(−r−s+2)⊗OP(V∨)(2)→ 0since −r − s+ 2 = −N + k + 1, by projection formula, we havep∗(OD ⊗OP(CN/V)(−r − s+ 2)⊗OP(V∨)(2)) = 0.So p∗[(V ′′/V ′′′)3 ⊗ (V ′/V)r+s−3] ∼= p∗[(V ′′/V ′′′)2 ⊗ (V ′/V)r+s−2].Then the rest is the same as the first case, and we would gett∗p∗[(V ′′/V ′′′)2 ⊗ (V ′/V)r+s−2] ∼= t∗p∗[(V ′′/V ′′′)3 ⊗ (V ′/V)r+s−3] ∼= ... ∼= (Fr+s ∗ E)1(k,N−k).Thus we end up with two exact triangles→ t∗p∗[V ′′/V ′′′ ⊗ (V ′/V)r+s−1]→ (E ∗ Fr+s)1(k,N−k) → ∆∗CN/V ⊗ det(CN/V)[1 + k −N ]→→ (Fr+s ∗ E)1(k,N−k) → t∗p∗[V ′′/V ′′′ ⊗ (V ′/V)r+s−1]→ ∆∗V ⊗ det(CN/V)[1 + k −N ]→ .Denoting the two maps by f : t∗p∗[V ′′/V ′′′ ⊗ (V ′/V)r+s−1] → (E ∗ Fr+s)1(k,N−k) and g : (Fr+s ∗E)1(k,N−k) → t∗p∗[V ′′/V ′′′ ⊗ (V ′/V)r+s−1], then we have a map f ◦ g : (Fr+s ∗ E)1(k,N−k) → (E ∗Fr+s)1(k,N−k) in Db(G(k,N)×G(k,N)).Any morphism in a triangulated category can be completed into a exact triangle→ (Fr+s ∗ E)1(k,N−k) f◦g−−→ (E ∗ Fr+s)1(k,N−k) → Cone(f ◦ g)→ .36So we have to determine Cone(f ◦ g). Using the axioms for triangulated category, we have the followingexact triangle→ Cone(g) = ∆∗V⊗det(CN/V)[1+k−N ]→ Cone(f◦g)→ Cone(f) = ∆∗CN/V⊗det(CN/V)[1+k−N ]→ .(5.2.23)Recalling that in a triangulated category D, if we have two objects A, C ∈ ob(D) and an element γ ∈Ext1D(A,C) = HomD(A,C[1]), then we can complete it into an exact triangleAγ−→ C[1]→ Cone(γ).Thus we get an exact triangleC → Cone(γ)[−1]→ Aand we say that Cone(γ)[−1] is determined by the element γ ∈ Ext1D(A,C).Thus we know that Cone(f ◦ g) is given by an element in Ext1G(k,N)×G(k,N)(∆∗(CN/V ⊗ det(CN/V)[1 +k −N ]),∆∗(V ⊗ det(CN/V)[1 + k −N ])) and it is easy to calculate thatExt1G(k,N)×G(k,N)(∆∗(CN/V ⊗ det(CN/V)[1 + k −N ]),∆∗(V ⊗ det(CN/V)[1 + k −N ]))∼= Ext1∆(CN/V,V)⊕End(Ω∆).Remark 5.2.7. By Lemma 5.2.9, we know that End(Ω∆) ∼= C.We show the following,Lemma 5.2.8. Cone(f ◦ g) is given by the nonzero element (0, id) ∈ Ext1∆(CN/V,V)⊕End(Ω∆).Proof. Note that we have the following short exact sequence on X ,0→ (V ′′/V ′′′)N−k+1 → (V ′/V)N−k+1 → O(N−k+1)D ⊗ (V ′/V)N−k+1 → 0.Applying the derived pushforward p∗, we get the following exact triangle on Yp∗((V ′′/V ′′′)N−k+1)→ p∗((V ′/V)N−k+1)→ p∗(O(N−k+1)D ⊗ (V ′/V)N−k+1)Then applying t∗, we get(Fr+s ∗ E)1(k,N−k) → (E ∗ Fr+s)1(k,N−k) → t∗p∗(O(N−k+1)D ⊗ (V ′/V)N−k+1)37So we conclude that Cone(f ◦ g) ∼= t∗p∗(O(N−k+1)D ⊗ (V ′/V)N−k+1).For each n ≥ 1, we have the short exact sequence on X0→ (V ′′/V ′′′)n → (V ′/V)n → OnD ⊗ (V ′/V)n → 0.Assuming n ≥ 2, then tensoring (5.2.19) by (V ′′/V ′′′)n−1, we get→ (V ′′/V ′′′)n → V ′/V ⊗ (V ′′/V ′′′)n−1 → OD ⊗ V ′/V ⊗ (V/V ′′′)n−1 → .Also, tensoring the n− 1 term by (V ′/V), we get→ (V ′′/V ′′′)n−1 ⊗ V ′/V → (V ′/V)n → O(n−1)D ⊗ (V ′/V)n → .Those fit together to form a diagram of exact triangles(V ′′/V ′′′)nid// V ′/V ⊗ (V ′′/V ′′′)n−1// OD ⊗ V ′/V ⊗ (V/V ′′′)n−1(V ′′/V ′′′)n// (V ′/V)n// OnD ⊗ (V ′/V)n0 // O(n−1)D ⊗ (V ′/V)n // O(n−1)D ⊗ (V ′/V)nSo we get the exact triangle→ OD ⊗ V ′/V ⊗ (V/V ′′′)n−1 → OnD ⊗ (V ′/V)n → O(n−1)D ⊗ (V ′/V)n →for all n ≥ 2.In particular, when n = 2, we have→ OD ⊗ V ′/V ⊗ V/V ′′′ → O2D ⊗ (V ′/V)2 → OD ⊗ (V ′/V)2 →tensoring it with (V ′/V)N−k−1 we get→ OD ⊗ (V ′/V)N−k ⊗ V/V ′′′ → O2D ⊗ (V ′/V)N−k+1 → OD ⊗ (V ′/V)N−k+1 → .38Applying t∗p∗, we get∆∗V ⊗ det(CN/V)[1 + k −N ]→ t∗p∗(O2D ⊗ (V ′/V)N−k+1)→ ∆∗CN/V ⊗ det(CN/V)[1 + k −N ]→ .Comparing this to the exact triangle (5.2.23), we getCone(f ◦ g) ∼= t∗p∗(O(N−k+1)D ⊗ (V ′/V)N−k+1) ∼= i∗p∗(O2D ⊗ (V ′/V)N−k+1)Denoting i′ : D ↪→ X to be the inclusion of the divisor D. Then we have O2D ⊗ (V ′/V)N−k+1 is given byan element inExt1X(i′∗((V ′/V)N−k+1), i′∗((V ′/V)N−k ⊗ V/V ′′′)) ∼= Ext1X(i′∗(V ′/V), i′∗(V/V ′′′))∼= Ext1D(V ′/V,V/V ′′′)⊕Hom(N∨D/X ⊗ V ′/V,V/V ′′′)Since N∨D/X ∼= OD(−D) ∼= (V/V ′′′) ⊗ (V ′/V)−1, we conclude that Hom(N∨D/X ⊗ V ′/V,V/V ′′′) ∼=End(OD) and O2D ⊗ (V ′/V)N−k+1 is given by (0, id) ∈ Ext1D(V ′/V,V/V ′′′)⊕End(OD).Thus Cone(f ◦ g) ∼= i∗p∗(O2D ⊗ (V ′/V)N−k+1) is given by (0, id) ∈ Ext1∆(CN/V,V)⊕End(Ω∆).So we obtain the following exact triangle→ (Fr+s ∗ E)1(k,N−k) → (E ∗ Fr+s)1(k,N−k) → t∗p∗(O2D ⊗ (V ′/V)N−k+1)→ .Note that we have the kernel Ψ+1(k,N−k) is ∆∗ det(CN/V)[1 + k − N ], and we have the following exacttriangle∆∗V ⊗ det(CN/V)[1 + k −N ]→ t∗p∗(O2D ⊗ (V ′/V)N−k+1)→ ∆∗CN/V ⊗ det(CN/V)[1 + k −N ]→ .Taking convolution with the kernel ∆∗ det(CN/V)−1[−1− k +N ], we get→ ∆∗V → pi13∗{pi∗12t∗p∗(O2D ⊗ (V ′/V)N−k+1)⊗ pi∗23∆∗(det(CN/V)−1)}[−1− k +N ]→ ∆∗CN/V →and the middle term is the kernel H11(k,N−k). Thus we show that t∗p∗(O2D ⊗ (V ′/V)N−k+1) is the kernel(Ψ+ ∗ H1)1(k,N−k) and prove the third case.Finally, we prove relation (13a), (14a). Before we prove it, we prove a lemma that will be used in the proof.39Lemma 5.2.9. End(ΩG(k,N)) ∼= C.Proof. Instead of proving End(ΩG(k,N)) ∼= C, we prove that End(TG(k,N)) ∼= C, where TG(k,N) is the tangentbundle.By Theorem 1.2.9 in Chapter 2 of [OSS], it is enough to show that TG(k,N) is a stable bundle. Also, by [L],we have existence of a Kahler-Einstein metric implies that the tangent bundle is stable.By [AP], we know that all homogeneous varieties are Kahler-Einstein, so we are done.Theorem 5.2.10. We have the following exact triangles→ (H1 ∗ Er)1(k,N−k) → (Er ∗ H1)1(k,N−k) → (Er+1⊕Er+1[1])1(k,N−k) →→ (Er ∗ H−1)1(k,N−k) → (H−1 ∗ Er)1(k,N−k) → (Er−1⊕Er−1[1])1(k,N−k) →→ (Fs ∗ H1)1(k,N−k) → (H1 ∗ Fs)1(k,N−k) → (Fs+1⊕Fs+1[1])1(k,N−k) →→ (H−1 ∗ Fs)1(k,N−k) → (Fs ∗ H−1)1(k,N−k) → (Fs−1⊕Fs−1[1])1(k,N−k) →for all r, s such that the relations in Definition 3.0.1 make sense.Proof. It suffices to prove the first case, the others are similar. Also, it is sufficient to do the case where r = 0.We calculate (E ∗ H1)1(k,N−k) first, which is pi13∗(pi∗12(H11(k,N−k))⊗ pi∗23(E1(k,N−k))).From the proof in Proposition 5.2.5, we have H11(k,N−k) ∈ Db(G(k,N)×G(k,N)) is determined by thefollowing exact triangle→ ∆∗(V)→ H11(k,N−k) → ∆∗(CN/V)→ .Moreover, it is determined by the element(0, id) ∈ Ext1G(k,N)×G(k,N)(∆∗(CN/V),∆∗(V)) ∼= Ext1G(k,N)(CN/V,V)⊕End(ΩG(k,N)).Basically, to understand (E ∗ H1)1(k,N−k), what we have to do is keep tracking the element (0, id) duringthe process of convolution. From Chapter 4, we assume all the functors in the FM transform are derived, so wedo not need to worry about the issue of exactness after applying each functor.For the first step, we have the following exact triangle→ pi∗12∆∗(V)→ pi∗12H11(k,N−k) → pi∗12∆∗(CN/V)→and pi∗12H11(k,N−k) is given by an element in Ext1G(k,N)×G(k,N)×G(k−1,N)(pi∗12∆∗(CN/V), pi∗12∆∗(V)).40By using adjunction, it is easy to calculate thatExt1G(k,N)×G(k,N)×G(k−1,N)(pi∗12∆∗(CN/V), pi∗12∆∗(V)) ∼= Ext1G(k,N)(CN/V,V)⊕End(ΩG(k,N)).(5.2.24)So pi∗12H11(k,N−k) is still given by (0, id) ∈ Ext1G(k,N)(CN/V,V)⊕End(ΩG(k,N)).Next, the kernel E1(k,N−k) is ι∗OFl(k−1,k). Then we tensor the whole exact triangle by pi∗23ι∗OFl(k−1,k),and we get the following exact triangle→ pi∗12∆∗(V)⊗pi∗23ι∗OFl(k−1,k) → pi∗12H11(k,N−k)⊗pi∗23ι∗OFl(k−1,k) → pi∗12∆∗(CN/V)⊗pi∗23ι∗OFl(k−1,k) → .(5.2.25)So pi∗12H11(k,N−k) ⊗ pi∗23ι∗OFl(k−1,k) is given by an element inExt1G(k,N)×G(k,N)×G(k−1,N)(pi∗12∆∗(CN/V)⊗ pi∗23ι∗OFl(k−1,k), pi∗12∆∗(V)⊗ pi∗23ι∗OFl(k−1,k)). (5.2.26)We have the following commutative diagramG(k,N)×G(k − 1, N) G(k,N)×G(k,N)×G(k − 1, N)G(k,N) G(k,N)×G(k,N)∆×idpi1 pi12∆(5.2.27)Using base change, we have pi∗12∆∗ ∼= (∆× id)∗pi∗1 . So (5.2.26) becomesExt1((∆× id)∗pi∗1(CN/V)⊗ pi∗23ι∗OFl(k−1,k), (∆× id)∗pi∗1(V)⊗ pi∗23ι∗OFl(k−1,k)) (5.2.28)∼= Ext1((∆× id)∗(pi∗1(CN/V)⊗ (∆× id)∗pi∗23ι∗OFl(k−1,k)), (∆× id)∗(pi∗1(V)⊗ (∆× id)∗pi∗23ι∗OFl(k−1,k))).(5.2.29)Note that we have pi23 ◦ (∆× id) = id, using adjunction we get that (5.2.29) is isomorphic toExt1G(k,N)×G(k−1,N)((∆× id)∗(∆× id)∗(pi∗1(CN/V)⊗ ι∗OFl(k−1,k)), pi∗1(V)⊗ ι∗OFl(k−1,k)) (5.2.30)∼= Ext1(pi∗1(CN/V)⊗ ι∗OFl(k−1,k), pi∗1(V)⊗ ι∗OFl(k−1,k)) (5.2.31)⊕Hom(N∨∆×G(k−1,N)/G(k,N)×G(k,N)×G(k−1,N) ⊗ pi∗1(CN/V)⊗ ι∗OFl(k−1,k), pi∗1(V)⊗ ι∗OFl(k−1,k)).(5.2.32)41Here, applying adjunction again, it is easy to calculate thet (5.2.31) is isomorphic toExt1G(k,N)×G(k−1,N)(pi∗1(CN/V)⊗ ι∗OFl(k−1,k), pi∗1(V)⊗ ι∗OFl(k−1,k))∼= Ext1Fl(k−1,k)(ι∗pi∗1(CN/V), ι∗pi∗1(V))⊕Hom(ι∗pi∗1(CN/V)⊗N∨Fl(k−1,k)/G(k,N)×G(k−1,N), ι∗pi∗1(V)).Note that we haveN∨∆×G(k−1,N)/G(k,N)×G(k,N)×G(k−1,N) ∼= (CN/V)∨ ⊗ V, and N∨Fl(k−1,k)/G(k,N)×G(k−1,N) ∼= (CN/V)∨ ⊗ V ′.So, (5.2.32) becomesHom(N∨∆×G(k−1,N)/G(k,N)×G(k,N)×G(k−1,N) ⊗ pi∗1(CN/V)⊗ ι∗OFl(k−1,k), pi∗1(V)⊗ ι∗OFl(k−1,k))∼= Hom((CN/V)∨ ⊗ V, (CN/V)∨ ⊗ V).Similarly,Hom(ι∗pi∗1(CN/V)⊗N∨Fl(k−1,k)/G(k,N)×G(k−1,N), ι∗pi∗1(V)) ∼= Hom((CN/V)∨ ⊗ V ′, (CN/V)∨ ⊗ V)Combining them, we getExt1G(k,N)×G(k,N)×G(k−1,N)(pi∗12∆∗(CN/V)⊗ pi∗23ι∗OFl(k−1,k), pi∗12∆∗(V)⊗ pi∗23ι∗OFl(k−1,k)) (5.2.33)∼= Ext1Fl(k−1,k)(CN/V,V)⊕Hom((CN/V)∨ ⊗ V ′, (CN/V)∨ ⊗ V)⊕Hom((CN/V)∨ ⊗ V, (CN/V)∨ ⊗ V).(5.2.34)Let i : V ′ → V is the natural inclusion, and id⊗ i ∈ Hom((CN/V)∨ ⊗ V ′, (CN/V)∨ ⊗ V). Then we showthatLemma 5.2.11. The natural mapExt1(pi∗12∆∗(CN/V), pi∗12∆∗(V))→ Ext1(pi∗12∆∗(CN/V)⊗ pi∗23ι∗OFl(k−1,k), pi∗12∆∗(V)⊗ pi∗23ι∗OFl(k−1,k))maps (0, id) to (0, id⊗ i, id).This tells us that the exact triangle (5.2.25) is determined by the element(0, id⊗ i, id) ∈ Ext1(pi∗12∆∗(CN/V)⊗ pi∗23ι∗OFl(k−1,k), pi∗12∆∗(V)⊗ pi∗23ι∗OFl(k−1,k))42.Finally, we apply pi13∗ and get the following exact triangle→ pi13∗(pi∗12∆∗(V)⊗ pi∗23ι∗OFl(k−1,k))→ pi13∗(pi∗12H11(k,N−k) ⊗ pi∗23ι∗OFl(k−1,k)) (5.2.35)→ pi13∗(pi∗12∆∗(CN/V)⊗ pi∗23ι∗OFl(k−1,k))→ pi13∗(pi∗12∆∗(V)⊗ pi∗23ι∗OFl(k−1,k))[1]. (5.2.36)Using the base change, we havepi13∗(pi∗12∆∗(V)⊗ pi∗23ι∗OFl(k−1,k)) ∼= pi13∗((∆× id)∗pi∗1(V)⊗ pi∗23ι∗OFl(k−1,k))∼= pi13∗(∆× id)∗(pi∗1(V)⊗ (∆× id)∗pi∗23ι∗OFl(k−1,k)) ∼= pi∗1(V)⊗ ι∗OFl(k−1,k)since that pi13 ◦ (∆× id) = pi23 ◦ (∆× id) = id.Then the exact triangle (5.2.35), (5.2.36) becomes→ pi∗1(V)⊗ ι∗OFl(k−1,k) → (E ∗ H1)1(k,N−k) → pi∗1(CN/V)⊗ ι∗OFl(k−1,k) →and (E ∗ H1)1k is given by Ext1G(k,N)×G(k−1,N)(pi∗1(CN/V)⊗ ι∗OFl(k−1,k), pi∗1(V)⊗ ι∗OFl(k−1,k)).Via adjunction, it is easy to calculate thatExt1G(k,N)×G(k−1,N)(pi∗1(CN/V)⊗ ι∗OFl(k−1,k), pi∗1(V)⊗ ι∗OFl(k−1,k))∼= Ext1Fl(k−1,k)(CN/V,V)⊕Hom((CN/V)∨ ⊗ V ′, (CN/V)∨ ⊗ V).Then it is easy to see thatLemma 5.2.12. The natural mapExt1(pi∗12∆∗(CN/V)⊗ pi∗23ι∗OFl(k−1,k), pi∗12∆∗(V)⊗ pi∗23ι∗OFl(k−1,k))→ Ext1(pi∗1(CN/V)⊗ ι∗OFl(k−1,k), pi∗1(V)⊗ ι∗OFl(k−1,k)),maps (0, id⊗ i, id) to (0, id⊗ i).So (E∗H1)1(k,N−k) is determined by the element (0, id⊗i) ∈ Ext1Fl(k−1,k)(CN/V,V)⊕Hom((CN/V)∨⊗V ′, (CN/V)∨ ⊗ V).Next, we calculate (H1 ∗ E)1(k,N−k), which is pi13∗(pi∗12(E1(k,N−k))⊗ pi∗23(H11(k−1,N−k+1))).Similarly, the kernel H11(k−1,N−k+1) ∈ Db(G(k − 1, N) × G(k − 1, N)) is determined by the following43exact triangle→ ∆∗(V ′)→ H11(k−1,N−k+1) → ∆∗(CN/V ′)→and it is determined by the nonzero element(0, id) ∈ Ext1G(k−1,N)×G(k−1,N)(∆∗(CN/V ′),∆∗(V ′)) ∼= Ext1G(k−1,N)(CN/V ′,V ′)⊕End(ΩG(k−1,N)).Then we use the same method as above. Applying pi∗23, we get the following exact triangle→ pi∗23∆∗(V ′)→ pi∗23H11(k−1,N−k+1) → pi∗23∆∗(CN/V ′)→ (5.2.37)and similar calculation shows that pi∗23H11(k−1,N−k+1) is again given by (0, id) ∈ Ext1(CN/V ′,V ′)⊕End(ΩG(k−1,N)).Tensoring (5.2.37) by pi∗12ι∗OFl(k−1,k), we get the following exact triangle→ pi∗12ι∗OFl(k−1,k)⊗pi∗23∆∗(V ′)→ pi∗12ι∗OFl(k−1,k)⊗pi∗23H11(k−1,N−k+1) → pi∗12ι∗OFl(k−1,k)⊗pi∗23∆∗(CN/V ′)→ .(5.2.38)Similar calculation shows that pi∗12ι∗OFl(k−1,k) ⊗ pi∗23H11(k−1,N−k+1) is determined by an element inExt1G(k,N)×G(k−1,N)×G(k−1,N)(pi∗12ι∗OFl(k−1,k) ⊗ pi∗23∆∗(CN/V ′), pi∗12ι∗OFl(k−1,k) ⊗ pi∗23∆∗(V ′))∼= Ext1Fl(k−1,k)(CN/V,V)⊕Hom((CN/V)∨ ⊗ V ′, (CN/V ′)∨ ⊗ V ′)⊕Hom((CN/V ′)∨ ⊗ V ′, (CN/V ′)∨ ⊗ V ′).Let pi : CN/V ′ → CN/V be the natural projection, and pi∨ : (CN/V)∨ → (CN/V ′)∨ be the induced map.Then we show thatLemma 5.2.13. The natural mapExt1(pi∗23∆∗(CN/V ′), pi∗23∆∗(V ′))→ Ext1(pi∗12ι∗OFl(k−1,k)⊗pi∗23∆∗(CN/V ′), pi∗12ι∗OFl(k−1,k)⊗pi∗23∆∗(V ′))maps (0, id) to (0, pi∨ ⊗ id, id).This tells us that the exact triangle (5.2.38) is given by the element(0, pi∨ ⊗ id, id) ∈ Ext1(pi∗12ι∗OFl(k−1,k) ⊗ pi∗23∆∗(CN/V ′), pi∗12ι∗OFl(k−1,k) ⊗ pi∗23∆∗(V ′))44Finally, we apply pi13∗ to get the following exact triangle→ pi13∗(pi∗12ι∗OFl(k−1,k) ⊗ pi∗23∆∗(V ′))→ pi13∗(pi∗12ι∗OFl(k−1,k) ⊗ pi∗23H11(k−1,N−k+1))→ pi13∗(pi∗12ι∗OFl(k−1,k) ⊗ pi∗23∆∗(CN/V ′))→ .Using base change and similar calculations gives the exact triangle→ ι∗OFl(k−1,k) ⊗ pi∗2(V ′)→ (H1 ∗ E)1(k−1,N−k+1) → ι∗OFl(k−1,k) ⊗ pi∗2(CN/V ′)→and (H1 ∗ E)1(k,N−k) is determined by an element inExt1G(k,N)×G(k−1,N)(ι∗OFl(k−1,k) ⊗ pi∗2(CN/V ′), ι∗OFl(k−1,k))⊗ pi∗2(V ′))∼= Ext1Fl(k−1,k)(CN/V ′,V ′)⊕Hom((CN/V)∨ ⊗ V ′, (CN/V ′)∨ ⊗ V ′).We show thatLemma 5.2.14. The natural mapExt1(pi∗12ι∗OFl(k−1,k) ⊗ pi∗23∆∗(CN/V ′), pi∗12ι∗OFl(k−1,k) ⊗ pi∗23∆∗(V ′))→ Ext1(ι∗OFl(k−1,k) ⊗ pi∗2(CN/V ′), ι∗OFl(k−1,k))⊗ pi∗2(V ′))maps (0, pi∨ ⊗ id, id) to (0, pi∨ ⊗ id).So (H1∗E)1(k,N−k) is determined by the element (0, pi∨⊗id) ∈ Ext1Fl(k−1,k)(CN/V ′,V ′)⊕Hom((CN/V)∨⊗V ′, (CN/V ′)∨ ⊗ V ′).Then we get the following diagram of exact trianglesι∗OFl(k−1,k) ⊗ pi∗2(V ′) //i(H1 ∗ E)1(k,N−k) // ι∗OFl(k−1,k) ⊗ pi∗2(CN/V ′) //piι∗OFl(k−1,k) ⊗ pi∗2(V ′)[1]i[1]pi∗1(V)⊗ ι∗OFl(k−1,k) // (E ∗ H1)1(k,N−k) // pi∗1(CN/V)⊗ ι∗OFl(k−1,k) // pi∗1(V)⊗ ι∗OFl(k−1,k)[1]In order to get an induced map (H1 ∗ E)1(k,N−k) → (E ∗ H1)1(k,N−k), we have to show that the followingdiagram commutes.45ι∗OFl(k−1,k) ⊗ pi∗2(CN/V ′)(0,pi∨⊗id)//piι∗OFl(k−1,k) ⊗ pi∗2(V ′)[1]i[1]pi∗1(CN/V)⊗ ι∗OFl(k−1,k)(0,id⊗i)// pi∗1(V)⊗ ι∗OFl(k−1,k)[1].Both i[1] ◦ (0, pi∨ ⊗ id) and (0, id⊗ i) ◦ pi are maps inExt1G(k,N)×G(k−1,N)(ι∗OFl(k−1,k) ⊗ pi∗2(CN/V ′), pi∗1(V)⊗ ι∗OFl(k−1,k))∼= Ext1Fl(k−1,k)(CN/V ′,V)⊕Hom((CN/V)∨ ⊗ V ′, (CN/V ′)∨ ⊗ V).Since the first components of (0, pi∨ ⊗ id) and (0, id⊗ i) are zero, we havei[1] ◦ (0, pi∨ ⊗ id), (0, id⊗ i) ◦ pi ∈ Hom((CN/V)∨ ⊗ V ′, (CN/V ′)∨ ⊗ V).For i[1] ◦ (0, pi∨ ⊗ id), this map is defined as follows(CN/V)∨ ⊗ V ′ pi∨⊗id−−−−→ (CN/V ′)∨ ⊗ V ′ id⊗i−−−→ (CN/V ′)∨ ⊗ Vwhile for (0, id⊗ i) ◦ pi, it is defined as follows(CN/V)∨ ⊗ V ′ id⊗i−−−→ (CN/V)∨ ⊗ V pi∨⊗id−−−−→ (CN/V ′)∨ ⊗ V.Thus the diagram commutes, and we get an induced map γ : (H1 ∗ E)1(k,N−k) → (E ∗ H1)1(k,N−k). Theabove diagram becomesι∗(V ′)i// (H1 ∗ E)1k //γι∗(CN/V ′) //piι∗(V ′)[1]i[1]ι∗(V) // (E ∗ H1)1k // ι∗(CN/V) // ι∗(V)[1]46Completing left and right hand sides into exact triangles, we haveι∗(V ′)i// (H1 ∗ E)1k //γι∗(CN/V ′) //piι∗(V ′)[1]i[1]ι∗(V) //j(E ∗ H1)1k //ι∗(CN/V) //ι∗(V)[1]j[1]ι∗(V/V ′) // Cone(γ) // ι∗(V/V ′)[1] // ι∗(V/V ′)[1]Then we show thatCone(γ) ∼= ι∗(V/V ′)⊕ι∗(V/V ′)[1]. (5.2.39)This is the same as showing the map ι∗(V/V ′)[1]→ ι∗(V/V ′)[1] is zero. Note thatHom(ι∗(V/V ′)[1], ι∗(V/V ′)[1]) = Hom(ι∗(V/V ′), ι∗(V/V ′)) = Hom(OFl(k−1,k),OFl(k−1,k)) ∼= C.(5.2.40)The only maps ι∗(V/V ′)[1] → ι∗(V/V ′)[1] are either zero or isomorphism (identity up to a nonzero con-stant). Since the map ι∗(CN/V) → ι∗(V/V ′)[1] is nonzero, if we want to show that the map ι∗(V/V ′)[1] →ι∗(V/V ′)[1] is zero, by commutativity of the square, it suffices to show that the composition mapι∗(CN/V) (0,id⊗i)−−−−−→ ι∗(V)[1] j[1]−−→ ι∗(V/V ′)[1] (5.2.41)is zero.From above argument, this map precisely is(CN/V)∨ ⊗ V ′ id⊗i−−−→ (CN/V)∨ ⊗ V id⊗j−−−→ (CN/V ′)∨ ⊗ V/V ′ (5.2.42)which is zero since j ◦ i = 0.Thus we get Cone(γ) ∼= ι∗(V/V ′)⊕ι∗(V/V ′)[1]. Since that ι∗(V/V ′) is the kernel for E11(k,N−k), weconclude that there is an exact triangle→ (H1 ∗ E)1(k,N−k) → (E ∗ H1)1(k,N−k) → (E1 ⊕ E1[1])1(k,N−k) → .Finally, we prove relation (4).47Lemma 5.2.15.(H11(k,N−k))L ∼= 1(k,N−k)H−1 ∼= (H11(k,N−k))RProof. It suffices to prove the first, the other is similar.By definition, we have H11(k,N−k) = (Ψ+1(k,N−k))−1 ∗ t∗p∗(O2D ⊗ (V ′/V)N−k+1). Taking left adjoint,we get(H11(k,N−k))L = {(Ψ+1(k,N−k))−1 ∗ t∗p∗(O2D ⊗ (V ′/V)N−k+1)}L∼= (t∗p∗(O2D ⊗ (V ′/V)N−k+1))L ∗ ((Ψ+1(k,N−k))−1)LNote that since (Ψ+1(k,N−k))−1 is invertible, the left adjoint is isomorphic to its inverse, i.e. ((Ψ+1(k,N−k))−1)L ∼=Ψ+1(k,N−k).Next, we need to compute (t∗p∗(O2D ⊗ (V ′/V)N−k+1))L. By definition, we have(t∗p∗(O2D ⊗ (V ′/V)N−k+1))L ∼= (t∗p∗(O2D ⊗ (V ′/V)N−k+1))∨ ⊗ pi∗2ωG(k,N)[dimG(k,N)]. (5.2.43)Using Grothendieck-Verdier duality, we show that(t∗p∗(O2D⊗(V ′/V)N−k+1))∨ ∼= t∗p∗((O2D⊗(V ′/V)N−k+1)∨⊗ωX)⊗ω−1G(k,N)×G(k,N)[dimX−dimG(k,N)×G(k,N)].(5.2.44)We know that D = Fl(k − 1, k, k + 1) ⊂ X is a divisor, so dimX = k(N − k) +N − 1. Next, we have(O2D ⊗ (V ′/V)N−k+1)∨ ∼= (O2D)∨ ⊗ ((V ′/V)N−k+1)∨ ∼= (O2D)∨ ⊗ (V ′/V)−N+k−1 (5.2.45)To calculate (O2D)∨, we have the following exact triangle→ i∗OD ⊗OX(−D)→ O2D → i∗OD →taking dual, we get→ (i∗OD)∨ → (O2D)∨ → (i∗OD)∨ ⊗OX(D)→ .Since (i∗OD)∨ ∼= i∗OD ⊗OX(D)[−1], the exact triangle becomes→ i∗OD ⊗OX(D)[−1]→ (O2D)∨ → i∗OD ⊗OX(2D)[−1]→ .48We conclude that (O2D)∨ ∼= O2D ⊗OX(2D)[−1]. Thus (5.2.45) becomesO2D ⊗OX(2D)⊗ (V ′/V)−N+k−1[−1] (5.2.46)and (5.2.44) becomesp∗(O2D ⊗OX(2D)⊗ (V ′/V)−N+k−1 ⊗ ωX)⊗ ω−1G(k,N)×G(k,N)[−k(N − k) +N − 2]. (5.2.47)We already know that OX(D)|D ∼= (V/V ′′′)−1 ⊗ (V ′/V), and by an adjunction formula, we also haveωX |D ∼= ωD ⊗OD(−D) ∼= (V/V ′′′)⊗ (V ′/V)−1 ⊗ ωD.An easy calculation gives ωD ∼= (V/V ′′′)−k ⊗ (V ′/V)N−k ⊗ det(V)⊗ det(CN/V)−1 ⊗ ωG(k,N).So combining all of them we get(t∗p∗(O2D ⊗ (V ′/V)N−k+1))L∼= t∗p∗(O2D ⊗OX(2D)⊗ (V ′/V)−N+k−1 ⊗ ωX)⊗ ω−1G(k,N)×G(k,N) ⊗ pi∗2ωG(k,N)[dimG(k,N)− k(N − k) +N − 2]∼= t∗p∗(O2D ⊗ (V/V ′′′)−k−1 ⊗ det(V)⊗ det(CN/V)−1 ⊗ ωG(k,N))⊗ ω−1G(k,N)×G(k,N) ⊗ pi∗2ωG(k,N)[N − 2]∼= t∗p∗(O2D ⊗ (V/V ′′′)−k−1)⊗∆∗ det(V)⊗∆∗ det(CN/V)−1[N − 2] (by projection formula).Thus,(H11(k,N−k))L ∼= (t∗p∗(O2D ⊗ (V ′/V)N−k+1))L ∗Ψ+1(k,N−k)∼= pi13∗(pi∗12∆∗ det(CN/V)⊗ pi∗23(t∗p∗(O2D ⊗ (V/V ′′′)−k−1)⊗∆∗ det(V)⊗∆∗ det(CN/V)−1))[N − 2 + 1 + k −N ]∼= pi13∗(pi∗12∆∗ det(V)⊗ pi∗23(t∗p∗(O2D ⊗ (V/V ′′′)−k−1)))[k − 1]which is the kernel 1(k,N−k)H−1.Thus we prove Theorem 5.2.1, and we would prove the rest relations in the next section.5.3 The rest relationsBy all of the above work, we have proved Theorem 5.2.1. Now we move to the sln case.To prove Theorem 5.1.2, note that relation (2), (3) are obvious. Since we have already proved the sl2 case,many relations in Definition 3.0.1 are a direct generalization of the sl2 version. Finally, it is easy to checkrelations (9c), (10c), (11c), (12c), (13c), (14c), (15).49So it remains to check the relations (6), (7), (8), (9b), (10b), (11b), (12b), (13b), (14b).Relation (7) is easy to show since Ψ±i 1k are just line bundles. This means that we haveLemma 5.3.1. ((Ψ±i )±1 ∗ (Ψ±j )±1)1k ∼= ((Ψ±j )±1 ∗ (Ψ±i )±1)1k for all i, j.The next is relation (8).Lemma 5.3.2. (Hi,±1 ∗Ψ±j )1k ∼= (Ψ±j ∗ Hi,±1)1k for all i, j.Proof. It suffices to prove the first case, the others are similar.From the proof of Proposition 5.2.5, we know that the kernel Hi,11k is determined by the following exacttriangle→ ∆∗Vi/Vi−1 → Hi,11k → ∆∗Vi+1/Vi → .More precisely, it is determined by the element(0, id) ∈ Ext1Y (k)×Y (k)(∆∗(Vi+1/Vi),∆∗(Vi/Vi−1)) ∼= Ext1Y (k)(Vi+1/Vi,Vi/Vi−1)⊕End(ΩY (k))By Definition 3.0.1, the kernel for Ψ+j 1k is ∆∗det(Vj+1/Vj)[1 − kj+1]. Taking convolution, we get thefollowing exact triangle→ ∆∗(Vi/Vi−1⊗ det(Vj+1/Vj))[1− kj+1]→ (Ψ+j ∗Hi,1)1k → ∆∗(Vi+1/Vi)⊗ det(Vj+1/Vj))[1− kj+1]→and→ ∆∗(Vi/Vi−1⊗det(Vj+1/Vj))[1−kj+1]→ (Hi,1 ∗Ψ+j )1k → ∆∗(Vi+1/Vi)⊗det(Vj+1/Vj))[1−kj+1]→ .Since tensoring a line bundle does not change the class that determines the exact triangle, the above two exacttriangles are still determined by the element (0, id) ∈ Ext1Y (k)×Y (k)(∆∗(Vi+1/Vi),∆∗(Vi/Vi−1)).Thus we obtain the isomorphism (Hi,1 ∗Ψ+j )1k ∼= (Ψ+j ∗ Hi,1)1k.Using the same argument in the proof of Lemma 5.3.2, we can prove relation (6), which is the following.Lemma 5.3.3. (Hi,±1 ∗ Hj,±1)1k ∼= (Hj,±1 ∗ Hi,±1)1k for all i, j.Relations (9b) and (10b).50Lemma 5.3.4. We have the following exact triangle→ (Ei+1,s ∗ Ei,r+1)1k → (Ei+1,s+1 ∗ Ei,r)1k → (Ei,r ∗ Ei+1,s+1)1k →→ (Fi,r+1 ∗ Fi+1,s)1k → (Fi,r ∗ Fi+1,s+1)1k → (Fi+1,s+1 ∗ Fi,r)1k → .Proof. It suffices to prove the first case, the other is similar.A simple calculation gives that (Ei+1,s+1 ∗ Ei,r)1k is given byi1∗(Vi/V ′′i )r ⊗ (Vi+1/V ′′i+1)s+1and the kernel (Ei,r ∗ Ei+1,s+1)1k is given byi2∗(Vi/V ′′i )r ⊗ (Vi+1/V ′′i+1)s+1.Here W 1,1i+1,i(k) ⊂ Y (k)× Y (k + αi + αi+1) and W 1,1i,i+1(k) ⊂ Y (k)× Y (k + αi + αi+1) are defined byW 1,1i+1,i(k) = {(V•, V ′′• )|V ′′i ⊂ Vi, V ′′i+1 ⊂ Vi+1, Vj = V ′′j for j 6= i, i+ 1}W 1,1i,i+1(k) = {(V•, V ′′• )|V ′′i ⊂ Vi ⊂ V ′′i+1 ⊂ Vi+1, Vj = V ′′j for j 6= i, i+ 1}and i1 : W1,1i+1,i(k)→ Y (k)×Y (k+αi+αi+1), i2 : W 1,1i,i+1(k)→ Y (k)×Y (k+αi+αi+1) are the inclusions.Note that we have W 1,1i,i+1(k) ⊂ W 1,1i+1,i(k) is a divisor that cut out by the natural section of the line bundleHom(Vi/V ′′i ,Vi+1/V ′′i+1). This implies that OW 1,1i+1,i(k)(W1,1i,i+1(k))∼= (Vi/V ′′i )∨ ⊗ Vi+1/V ′′i+1.From the divisor short exact sequence0→ OW 1,1i+1,i(k)(−W 1,1i,i+1(k)) ∼= Vi/V ′′i ⊗ (Vi+1/V ′′i+1)−1 → OW 1,1i+1,i(k) → OW 1,1i,i+1(k) → 0Tensoring with (Vi/V ′′i )r ⊗ (Vi+1/V ′′i+1)s+1 we get0→ (Vi/V ′′i )r+1⊗(Vi+1/V ′′i+1)s → (Vi/V ′′i )r⊗(Vi+1/V ′′i+1)s+1 → OW 1,1i,i+1(k)⊗(Vi/V′′i )r⊗(Vi+1/V ′′i+1)s+1 → 0Applying i1∗ and by comparing kernels, we get→ (Ei+1,s ∗ Ei,r+1)1k → (Ei+1,s+1 ∗ Ei,r)1k → (Ei,r ∗ Ei+1,s+1)1k →51Next, we check relations (11b), (12b).Lemma 5.3.5.(Ψ±i ∗ Ei±1,r)1k ∼= (Ei±1,r−1 ∗Ψ±i )1k[±1] (Ψ±i ∗ Fi±1,r)1k ∼= (Fi±1,r+1 ∗Ψ±i )1k[∓1](Ψ±i ∗ Ei∓1,r)1k ∼= (Ei∓1,r ∗Ψ±i )1k (Ψ±i ∗ Fi∓1,r)1k ∼= (Fi∓1,r ∗Ψ±i )1kProof. It suffices to prove the first case, the others are similar.An easy calculation gives that (Ψ+i ∗ Ei+1,r)1k is given byι∗(Vi+1/V ′i+1)r ⊗ det(V ′i+1/Vi)[2− ki+1]Similarly, (Ei+1,r−1 ∗Ψ+i )1k is given byι∗(Vi+1/V ′i+1)r−1 ⊗ det(Vi+1/Vi)[1− ki+1]Here ι : W 1i+1(k)→ Y (k)× Y (k + αi+1) is the inclusion. Note that we have the short exact sequence0→ V ′i+1/Vi → Vi+1/Vi → Vi+1/V ′i+1 → 0which implies that det(V ′i+1/Vi)⊗ Vi+1/V ′i+1 ∼= det(Vi+1/Vi).Combining the above, we getι∗(Vi+1/V ′i+1)r ⊗ det(V ′i+1/V ′i)[2− ki+1] ∼= ι∗(Vi+1/V ′i+1)r−1 ⊗ det(Vi+1/Vi)[2− ki+1]Finally, we prove relation (13b), (14b).52Lemma 5.3.6. We have the following exact triangles→ (Hi,1 ∗ Ei+1,r)1k → (Ei+1,r ∗ Hi,1)1k → Ei+1,r+11k →→ Ei−1,r+11k → (Hi,1 ∗ Ei−1,r)1k → (Ei−1,r ∗ Hi,1)1k →→ Ei+1,r−11k → (Ei+1,r ∗ Hi,−1)1k → (Hi,−1 ∗ Ei+1,r)1k →→ (Ei−1,r ∗ Hi,−1)1k → (Hi,1 ∗ Ei−1,r)1k → Ei−1,r−11k →→ (Fi+1,r ∗ Hi,1)1k → (Hi,1 ∗ Fi+1,r)1k → Fi+1,r+11k →→ Fi−1,r+11k → (Fi−1,r ∗ Hi,1)1k → (Hi,1 ∗ Fi−1,r)1k →→ Fi+1,r−11k → (Hi,−1 ∗ Fi+1,r)1k → (Fi+1,r ∗ Hi,−1)1k →→ (Hi,−1 ∗ Ei−1,r)1k → (Ei−1,r ∗ Hi,−1)1k → Fi−1,r−11k →Proof. It suffices to prove the first case, the others are similar. We know that the kernelHi,11k is determined bythe exact triangle→ ∆∗Vi/Vi−1 → Hi,11k → ∆∗Vi+1/Vi → .We would use this exact triangle to calculate the convolution of kernels.First, we calculate the convolution (Ei+1,r ∗ Hi,1)1k, which is in the following exact triangle→ Ei+1,r1k ∗ (∆∗Vi/Vi−1)→ (Ei+1,r ∗ Hi,1)1k → Ei+1,r1k ∗ (∆∗Vi+1/Vi)→We denote the larger spaceY (k)× Y (k)× Y (k + αi+1) = {(V•, V ′′• , V ′•)}and the kernel Ei+1,r1k is ι∗(V ′′i+1/V ′i+1)r, where W 1i+1(k) was defined in Section 5.1.An easy calculation gives us the following exact triangleι∗Vi/Vi−1 ⊗ (Vi+1/V ′i+1)r → (Ei+1,r ∗ Hi,1)1k → ι∗Vi+1/Vi ⊗ (Vi+1/V ′i+1)rNext, we have the kernelHi,11k+αi+1 ∈ Db(Y (k+αi+1)×Y (k+αi+1)) is determined by the exact triangle→ ∆∗V ′i/V ′i−1 → Hi,11k+αi+1 → ∆∗V ′i+1/V ′i → .53We calculate the convolution (Hi,1 ∗ Ei+1,r)1k, which is in the following exact triangle→ (∆∗V ′i/V ′i−1) ∗ Ei+1,r1k → (Hi,1 ∗ Ei+1,r)1k → (∆∗V ′i+1/V ′i) ∗ Ei+1,r1k → .We denote the larger spaceY (k)× Y (k + αi+1)× Y (k + αi+1) = {(V•, V ′′′• , V ′•)}and the kernel Ei+1,r1k is given by OW 1i+1(k) ⊗ (Vi+1/V ′′′i+1)r.An easy calculation gives us the following exact triangleι∗Vi/Vi−1 ⊗ (Vi+1/V ′i+1)r → (Hi,1 ∗ Ei+1,r)1k → ι∗V ′i+1/Vi ⊗ (Vi+1/V ′i+1)r.Now, on W 1i+1(k) = {(V•, V ′•)|V ∈ Y (k), V ′• ∈ Y (k + αi+1), Vj = V ′j for j 6= i + 1, and V ′i+1 ⊂ Vi+1},we have the following short exact sequence0→ V ′i+1/Vi → Vi+1/Vi → Vi+1/V ′i+1 → 0Then we have the following diagram of exact trianglesι∗Vi/Vi−1 ⊗ (Vi+1/V ′i+1)r (Hi,1 ∗ Ei+1,r)1k ι∗V ′i+1/Vi ⊗ (Vi+1/V ′i+1)rι∗Vi/Vi−1 ⊗ (Vi+1/V ′i+1)r (Ei+1,r ∗ Hi,1)1k ι∗Vi+1/Vi ⊗ (Vi+1/V ′i+1)rIn order to get an induced map (Hi,1 ∗ Ei+1,r)1k → (Ei+1,r ∗ Hi,1)1k, we have to show that the followingdiagram commutes.ι∗V ′i+1/Vi ⊗ (Vi+1/V ′i+1)r //ι∗Vi/Vi−1 ⊗ (Vi+1/V ′i+1)r[1]ι∗Vi+1/Vi ⊗ (Vi+1/V ′i+1)r // ι∗Vi/Vi−1 ⊗ (Vi+1/V ′i+1)r[1].This can be done by the same argument as in the proof of Theorem 5.2.10. Completing the diagram to exacttriangles, we get54ι∗Vi/Vi−1 ⊗ (Vi+1/V ′i+1)r (Hi,1 ∗ Ei+1,r)1k ι∗V ′i+1/Vi ⊗ (Vi+1/V ′i+1)rι∗Vi/Vi−1 ⊗ (Vi+1/V ′i+1)r (Ei+1,r ∗ Hi,1)1k ι∗Vi+1/Vi ⊗ (Vi+1/V ′i+1)r0 ι∗(Vi+1/V ′i+1)r+1 ι∗(Vi+1/V ′i+1)r+1and ι∗(Vi+1/V ′i+1)r+1 is the kernel for Ei+1,r+11k.Thus we get the exact triangle→ (Hi,1 ∗ Ei+1,r)1k → (Ei+1,r ∗ Hi,1)1k → Ei+1,r+11k →Combing the above results in this section and Section 5.2, we prove Theorem 5.1.2.55Chapter 6q = 0 affine Hecke algebras6.1 Definition and actionIn this section, we introduce the q = 0 affine Hecke algebras (of type A) and explain the relation to shifted q = 0affine algebras that defined above. We recall the definition of affine Hecke algebras (of type A).Definition 6.1.1. Let q ∈ C∗ with q 6= 1. The affine Hecke algebraHN (q) is defined to be the unital associativeC-algebra generated by the elements T1, ..., TN−1, X±11 , ..., X±1N subject to the following relations(Ti − q)(Ti + 1) = 0 (H1)TiTj = TjTi, if |i− j| ≥ 2 (H2)TiTjTi = TjTiTj , if |i− j| = 1 (H3)XiX−1i = X−1i Xi = 1 (H4)XiXj = XjXi, for all i, j (H5)TiXj = XjTi if j 6= i, i+ 1 (H6)TiXi = Xi+1Ti − (q − 1)Xi+1 (H7)TiXi+1 = XiTi + (q − 1)Xi+1 (H8)For q = 0 affine Hecke algebra, we take q = 0 in Definition 6.1.1, then the first relation (H1) becomesTi(Ti + 1) = 0. We replace Ti by −Ti, then it becomes T 2i = Ti. The relations H7 and H8 become −TiXi =−Xi+1Ti +Xi+1 and −TiXi+1 = −XiTi −Xi+1, respectively. Then we obtain the following definition.56Definition 6.1.2. The q = 0 affine Hecke algebra HN (0) is defined to be the unital associative C-algebragenerated by the elements T1, ..., TN−1, X±11 , ..., X±1N subject to the following relationsT 2i = Ti (H01)TiTj = TjTi, if |i− j| ≥ 2 (H02)TiTjTi = TjTiTj , if |i− j| = 1 (H03)XiX−1i = X−1i Xi = 1 (H04)XiXj = XjXi, for all i, j (H05)TiXj = XjTi if j 6= i, i+ 1 (H06)Xi+1Ti = TiXi +Xi+1 (H07)XiTi = TiXi+1 −Xi+1 (H08)We define the full flag variety to beFl := {0 ⊂ V1 ⊂ V2 ⊂ ... ⊂ VN = CN | dimVi = i for all i}and similarly the partial flag varietiesFlk := {0 ⊂ V1 ⊂ V2 ⊂ ...Vk−1 ⊂ Vk+1 ⊂ ...VN = CN | dimVi = i for i 6= k}by forgetting the kth vector space Vk.Then there is an action of the q = 0 affine Hecke algebra HN (0) on the K(Fl), which is the K-theoryof Fl. The construction is as follows. On Fl, we still denote Vi to be the tautological bundle of rank i. Letai = [Vi/Vi−1] be the class of the tautological line bundle Vi/Vi−1 in the K-theory (Grothendieck group) of Fl,where 1 ≤ i ≤ N .We have the Grothendieck group isK(Fl) = C[a±1 , ..., a±N ]/〈ei −(Ni)〉where 〈ei−(Ni)〉 is the ideal generated by {ei−(Ni )}Ni=1 and ei is the i-th symmetric polynomial in a1, a2, ..., aN .We have the natural maps pik : Fl → Flk which is just forgetting the k-dimensional vector spaces Vk forall 1 ≤ k ≤ N − 1. Those maps induce maps between the Grothendieck groups, they are the pushforwards57pik∗ : K(Fl) → K(Flk) and pullbacks pi∗k : K(Flk) → K(Fl), 1 ≤ k ≤ N − 1. Then we define the operatorTk := pi∗kpik∗ : K(Fl) → K(Fl) for all 1 ≤ k ≤ N − 1. Also, we define the operators Xi : K(Fl) → K(Fl)to be multiplication by ak for all 1 ≤ i ≤ N .It is easy to check that those operators Ti, X±1j defined in such way satisfy the relations of the q = 0 affineHecke algebra. Thus we get an action of HN (0) on K(Fl). Moreover, if we denote sk to be the permutationoperator for ak and ak+1, for 1 ≤ k ≤ N − 1, then Tk is given byTk(f) =ak+1f − aksk(f)ak+1 − akfor all f ∈ K(Fl) and 1 ≤ k ≤ N − 1.We would lift this action to the derived category of coherent sheaves on Fl, i.e. Db(Fl). This meansthat we need to construct functors Ti : Db(Fl) → Db(Fl) and Xj : Db(Fl) → Db(Fl) satisfy the relations incategorical setting. Again, we would use the language of FM transforms from Chapter 4 to define our categoricalaction.To define the functors Ti via using FM kernels, from the definition of Ti in K-theory, Ti = pi∗i pii∗ , we knowthat they are defined by using the forgetting map pii : Fl → Fli. Since pii also induce functors on derivedcategories of coherent sheaves, i.e. the derived pushforward pii∗ : Db(Fl) → Db(Fli) and the derived pullbackpi∗i : Db(Fli) → Db(Fl), naturally we can define the functors Ti := pi∗i pii∗ : Db(Fl) → Db(Fl) for all1 ≤ i ≤ N − 1.We can write Ti in terms of FM transforms. Recall that for a morphism f : X → Y between smoothprojective varieties, the derived pushforward f∗ : Db(X) → Db(Y ) is isomorphic to the FM transform withkernel OΓf , where Γf = (id × f)(X) is the graph of f in X × Y . Similarly, the derived pullback f∗ isisomorphic to the FM transform with kernel O(f×id)(X) in Db(Y × X). So the functor pii∗ is given by thekernel O(id×pii)(Fl), and the functor pi∗i is given by the kernel O(pii×id)(Fl), i.e. we have pii∗ ∼= ΦO(id×pii)(Fl) ,pi∗i ∼= ΦO(pii×id)(Fl) .So Ti ∼= ΦO(pii×id)(Fl) ◦ ΦO(id×pii)(Fl) . By Proposition 4.0.2, Ti is again a FM transform, moreover if we letTi to be the FM kernel of Ti, then Ti = O(pii×id)(Fl) ∗ O(id×pii)(Fl). To calculate it, by definition, we haveO(pii×id)(Fl) ∗ O(id×pii)(Fl) = pi13∗(pi∗12(O(id×pii)(Fl))⊗ pi∗23(O(pii×id)(Fl)))= pi13∗(O(id×pii)(Fl)×Fl ⊗OFl×(pii×id)(Fl))It is easy to calculate that the intersection of (id× pii)(Fl)× Fl and Fl × (pii × id)(Fl) in Fl × Fli × Fl58is smooth of expected codimension 2 dimFli. By Lemma 4.0.4, we getO(id×pii)(Fl)×Fl ⊗OFl×(pii×id)(Fl) ∼= O{(id×pii)(Fl)×Fl}∩{Fl×(pii×id)(Fl)}Let Fl×Fli Fl denote the fibred product, then pi13 maps the intersection (id×pii)(Fl)×Fl}∩{Fl× (pii×id)(Fl) into the fibred product Fl ×Fli Fl isomorphically. Thus pi13∗(O{(id×pii)(Fl)×Fl}∩{Fl×(pii×id)(Fl)}) ∼=OFl×FliFl, and Ti ∼= OFl×FliFl.Next, we define the functors Xj : Db(Fl) → Db(Fl), on K-theory, by multiplication by the elementaj = [Vj/Vj−1]. So it is natural to define Xj to be the functor that given by tensoring the line bundle Vj/Vj−1.Similarly, its FM kernel is given by Xj = ∆∗(Vj/Vj−1) ∈ Db(Fl × Fl), for all 1 ≤ j ≤ N where ∆ : Fl →Fl × Fl is the diagonal map. Also, we define X−1j to be the functor that given by tensoring the line bundle(Vj/Vj−1)−1, with the FM kernel given by X−1j = ∆∗((Vj/Vj−1)−1) ∈ Db(Fl × Fl).Theorem 6.1.3. There is a categorical action of the q = 0 affine Hecke algebra HN (0) on Db(Fl). Moreprecisely, if we define the FM kernels Ti = OFl×FliFl and Xj = ∆∗(Vj/Vj−1) for 1 ≤ i ≤ N − 1, 1 ≤ j ≤ N ,then we have the following relationsTi ∗ Ti ∼= Ti (6.1.1)Ti ∗ Tj ∼= Tj ∗ Ti, if |i− j| ≥ 2 (6.1.2)Ti ∗ Tj ∗ Ti ∼= Tj ∗ Ti ∗ Tj , if |i− j| = 1 (6.1.3)Xi ∗ X−1i ∼= X−1i ∗ Xi ∼= O∆ (6.1.4)Xi ∗ Xj ∼= Xj ∗ Xi, for all i, j (6.1.5)Ti ∗ Xj ∼= Xj ∗ Ti if j 6= i, i+ 1 (6.1.6)We have the following exact triangle→ Ti ∗ Xi → Xi+1 ∗ Ti → Xi+1 → (6.1.7)We have the following exact triangle→ Xi ∗ Ti → Ti ∗ Xi+1 → Xi+1 → (6.1.8)Instead of proving this theorem by direct computation of kernels, we can write the generators (or kernels)in terms of the generators (or kernels) of the q = 0 shifted affine algebra. Note that for the full flag variety, theweight is k = (1, 1, ..., 1), i.e. Fl = Fl(1,1,...,1)(CN ) = Y ((1, 1, ..., 1)) from the notation in Chapter 5, and sim-59ilarly for the partial flag varieties Fli by forgetting Vi, we have Fli = Fl(1,1,..,1)+αi(CN ) = Fl(1,1,..,1)−αi(CN ).By definition, we have the kernel Ei1(1,1,..,1) = ι∗OW 1i ((1,1,...,1)), whereW 1i ((1, 1, ..., 1)) := {(V•, V ′•) ∈ Fl × Fli|Vj = V ′j for j 6= i, and V ′i ⊂ Vi}Note that here V ′i = Vi−1, so the condition V′i ⊂ Vi automatically satisfy. Then W 1i ((1, 1, ..., 1)) = (id ×pii)(Fl), and we have Ei1(1,1,..,1) = ι∗O(id×pii)(Fl), which is the kernel for the pushforward functor pii∗. Usingthe same argument, we can show that Fi1(1,1,...,1) is also the kernel for pii∗ and Ei1(1,1,..,1)−αi ∼= Fi1(1,1,..,1)+αiare kernel for pi∗i .Since Ti := pi∗i pii∗, by above argument, the kernel Ti is isomorphic toTi ∼= (Ei ∗ Fi)1(1,1,...,1) ∼= (Fi ∗ Ei)1(1,1,...,1)Note that the second isomorphism can be seen from Proposition 5.2.5. For the kernel Xi, since k = (1, 1, ..., 1),it is easy to see that Xi ∼= Ψ+i−11(1,1,...,1) ∼= (Ψ−i )−11(1,1,...,1) for all 1 ≤ i ≤ NNow we already write Ti, Xj in terms of generators from q = 0 shifted affine algebra. To prove Theorem6.1.3, we need more relations to help.The first is the idempotent property when acting on the two end sides of weight spaces. It is easy to show bydirect computation.Lemma 6.1.4.(Fi ∗ Ei ∗ Fi ∗ Ei)1k+(ki−1)αi ∼= (Fi ∗ Ei)1k+(ki−1)αi(Ei ∗ Fi ∗ Ei ∗ Fi)1k−(ki+1−1)αi ∼= (Ei ∗ Fi)1k−(ki+1−1)αi(Ei ∗ Fi ∗ Ei ∗ Fi)1k+kiαi ∼= (Ei ∗ Fi)1k+kiαi(Fi ∗ Ei ∗ Fi ∗ Ei)1k−ki+1αi ∼= (Fi ∗ Ei)1k−ki+1αiThe next relation is the Serre relation. This can be deduced from the relations in the definition of categoricalaction, i.e. Definition 3.0.1.60Lemma 6.1.5.(Ei+1 ∗ Ei ∗ Ei+1)1k ∼= (Ei+1 ∗ Ei+1 ∗ Ei)1k(Ei ∗ Ei+1 ∗ Ei)1k ∼= (Ei+1 ∗ Ei ∗ Ei)1k(Fi+1 ∗ Fi ∗ Fi+1)1k ∼= (Fi ∗ Fi+1 ∗ Fi+1)1k(Fi ∗ Fi+1 ∗ Fi)1k ∼= (Fi ∗ Fi ∗ Fi+1)1kNow we prove Theorem 6.1.3.Proof. For relation (6.1.1), since Ti ∼= (Ei ∗ Fi)1(1,1,...,1) ∼= (Fi ∗ Ei)1(1,1,...,1), we can choose one of theisomorphism, say Ti ∼= (Ei ∗ Fi)1(1,1,...,1). Then we have to show that (Ei ∗ Fi ∗ Ei ∗ Fi)1(1,1,...,1) ∼= (Ei ∗Fi)1(1,1,...,1). But this is directly from Lemma 6.1.4.For relation (6.1.2), again, we use Ti ∼= (Ei ∗ Fi)1(1,1,...,1), then it follows from relation (13) in Definition3.0.1 of the categorical action.For relation (6.1.3), we prove the case for j = i + 1, the case for j = i − 1 is similar. We use Ti ∼=(Ei ∗ Fi)1(1,1,...,1), thenTi ∗ Ti+1 ∗ Ti ∼= (Ei ∗ Fi ∗ Ei+1 ∗ Fi+1 ∗ Ei ∗ Fi)1(1,1,...,1) ∼= (Ei ∗ Ei+1 ∗ Ei ∗ Fi ∗ Fi+1 ∗ Fi)1(1,1,...,1)∼= (Ei+1 ∗ Ei ∗ Ei ∗ Fi ∗ Fi ∗ Fi+1)1(1,1,...,1) ∼= (Ei+1 ∗ Ei ∗ Fi ∗ Ei ∗ Fi ∗ Fi+1)1(1,1,...,1) (by Lemma 6.1.5)∼= (Ei+1 ∗ Ei ∗ Fi ∗ Fi+1)1(1,1,...,1) ∼= (Fi ∗ Ei+1 ∗ Fi+1 ∗ Ei)1(1,1,...,1)∼= (Fi ∗ Ei+1 ∗ Fi+1 ∗ Ei+1 ∗ Fi+1 ∗ Ei)1(1,1,...,1) (by Lemma 6.1.4)∼= (Ei+1 ∗ Ei+1 ∗ Ei ∗ Fi ∗ Fi+1 ∗ Fi+1)1(1,1,...,1) ∼= (Ei+1 ∗ Ei ∗ Ei+1 ∗ Fi+1 ∗ Fi ∗ Fi+1)1(1,1,...,1) (by Lemma 6.1.5)∼= (Ei+1 ∗ Fi+1 ∗ Ei ∗ Fi ∗ Ei+1 ∗ Fi+1)1(1,1,...,1) ∼= Ti+1 ∗ Ti ∗ Ti+1For relations (6.1.4) and (6.1.5), they follow from the definition directly.For relation 6.1.6, let j 6= i, i + 1. Since Xj ∼= Ψ+j−11(1,1,...,1) and Ti ∼= (Ei ∗ Fi)1(1,1,...,1), it follows fromLemma 5.3.5.For relation (6.1.7) and (6.1.8), it suffices to prove one of them, say relation (6.1.7). By Proposition 5.2.5 orrelation (14)(c) in Definition 3.0.1 of the categorical action, we have the following exact triangle→ (Fi ∗ Ei,1)1(1,1,...,1) → (Ei,1 ∗ Fi)1(1,1,....,1) → Ψ+i 1(1,1,...,1) → (Fi ∗ Ei,1)1(1,1,...,1)[1]→Then note that for the kernel Ei,11(1,1,...,1), it is given by ι∗O(id×pii)(Fl) ⊗ Vi/Vi−1. So Ei,11(1,1,...,1) ∼=(Ei ∗ (Ψ−i )−1)1(1,1,...,1). Similarly, the kernel Ei,11(1,...,1,2,0,...,1), where 2 is at the ith position, is given by61ι∗O(id×pii)(Fl) ⊗ Vi+1/V ′i. So Ei,11(1,...,1,2,0,...,1) ∼= (Ψ+i ∗ Ei)1(1,...,1,2,0,...,1).So the exact triangle becomes→ (Fi ∗ Ei ∗ (Ψ−i )−1)1(1,1,...,1) → (Ψ+i ∗ Ei ∗ Fi)1(1,1,....,1) → Ψ+i 1(1,1,...,1) → (Fi ∗ Ei,1)1(1,1,...,1)[1]→Since Ti ∼= (Ei ∗ Fi)1(1,1,...,1) ∼= (Fi ∗ Ei)1(1,1,...,1) and Xi ∼= Ψ+i−11(1,1,...,1) ∼= (Ψ−i )−11(1,1,...,1), we canconclude that→ (Ti ∗ Xi)→ (Xi+1 ∗ Ti)→ Xi+1 → (Ti ∗ Xi)[1]→6.2 Relation to Demazure descentIn this section, we relate the action of q = 0 affine Hecke algebra that studied in Section 6.1 to the notionof Demazure descent, which is defined in [AK1]. They introduced the notion of Demazure descent data on atriangulated category and define the descent category for such data.Before we define the notion of Demazure descent, we fix some notations. Let G be a reductive algebraicgroup, and T be a Cartan subgroup. Choose a Borel subgroup T ⊂ B ⊂ G. Denote the set of simple roots by{α1, ..., αn}, and the simple reflection corresponding to a simple root αi is denoted by si. The Weyl group Wis the group that generated by those simple reflections.For an element w ∈ W , we denote the length l(w) to be the number of minimal expression of w via thegenerators {si}. We define the braid monoid Br+ of G to be the monoid with generator {Tw, w ∈ W} andrelationsTw1Tw2 = Tw1w2 if l(w1w2) = l(w1) + l(w2).Consider a triangulated category C.Definition 6.2.1. A weak braid monoid action on the category C is a collection of triangulated functorsDw : C → C, w ∈Wsatisfying braid monoid relations, i.e. for all w1, w2 ∈W there exist isomorphisms of functorsDw1 ◦Dw2 ∼= Dw1w2 if l(w1w2) = l(w1) + l(w2).A comonad structure (, η) on a endofunctor D : C → C is a structure of co-associative coalgebra on D inthe monoidal category of endofunctors. Here  : D → IC is the counit and η : D → D ◦D is the coproduct. A62comonad D is a co-projector if the coproduct map η : D → D ◦D is an isomorphism.Then the following is the definition of Demazure descent.Definition 6.2.2. A Demazure descent data on C is a weak braid monoid action {Dw} together with a co-projector structure (i, ηi) on the functor Dsi for every simple reflection si.The next is the definition of decent category.Definition 6.2.3. The descent category Desc(C, Dw,w∈W ) is the full subcategory of C with objects M such thatthe cone of i(M) : Dsi(M)→M is isomorphic to zero for all i.Then in [AK2], they define the quasi-coherent Hecke category and using the language of Fourier-Mukaitransformation to construct Demazure descent data on certain triangulated category. Fix a G-scheme X , wedenote QCoh(X) (resp. QCohG(X)) to be the categories of quasi-coherent sheaves (resp. G-equivariantquasi-coherent sheaves on X). Denote DQCoh(X) (resp. DQCohG(X)) to be the associated unboundedderived categories.Definition 6.2.4. The quasi-coherent Hecke category, denoted byQCHecke(G,B), is defined to be the monoidalcategory (DQCohG(G/B×G/B), ∗). Here ∗ is the convolution product of FM kernels that defined in Chapter4.Consider the category DQCohG(G/B × X), it is easy to see that the quasi-coherent Hecke categoryQCHecke(G,B) acts on it by the convolution product, i.e.QCHecke(G,B)×DQCohG(G/B ×X)→ DQCohG(G/B ×X)(F ,G) 7→ G ∗ FThere are special objects in the quasi-coherent Hecke category QCHecke(G,B) that can be used to definethe Demazure descent. For each simple root αi, we denote Pi to be the standard parabolic subgroup correspond-ing to αi and containing B. We define Rsi = G/B ×G/Pi G/B and ORsi ∈ DQCohG(G/B × G/B).More generally, for w ∈ W with a reduced expression w = si1si2 ...sik , where l(w) = k. We defineRw = G/B ×G/Pi1 G/B ×G/Pi2 G/B...×G/Pik G/B.Define the functor Dw : DQCohG(G/B ×X)→ DQCohG(G/B ×X), Dw(M) := M ∗ORw . Then thefollowing are the main results in [AK2].Theorem 6.2.5. (Theorem 5.5 in [AK2]) The functors {Dw} form Demazure data on DQCohG(G/B ×X)Theorem 6.2.6. (Theorem 6.1 in [AK2]) The descent category Desc(DQCohB(X), Dw, w ∈W ) is equivalentto DQCohG(X).63Now we back to our picture of the action of q = 0 affine Hecke algebra. Since our case is of type A, wehave G/B ×G/Pi G/B = Fl ×Fli Fl. So the kernel ORsi is precisely the kernel Ti in Section 6.1, if considerto be G-equivariantly. Similarly for ORw , which is the kernel Tw.We can see that the Demazure descent data on a triangulated category is part of the action of the q = 0 affineHecke algebra on the category. For those kernels Ti that defined in Section 6.1, it is easy to see that they forma braid monoid. As we proved in Theorem 6.1.3, the relation Ti ∗ Ti ∼= Ti implies that there is a co-projectorstructure on Ti for all i.Finally, we restrict to the bounded derived categories of (equivariant) coherent sheaves, which is denotedby DbCoh(.) or DbCohG(.). Similarly, we can also define the so called coherent Hecke category, denotedby CHecke(G,B), which is defined to be the monoidal category (DCohG(G/B × G/B), ∗). The followingcorollary is a direct consequence of Theorem 6.2.6 .Corollary 6.2.7. The descent category Desc(DbCohB(X), Dw, w ∈W ) is equivalent to DbCohG(X).If we take X = pt, then CohB(pt) ∼= Rep(B) and CohG(pt) ∼= Rep(G). This gives the following result,which is the main theorem in [AK1].Corollary 6.2.8. The descent category Desc(Db(Rep(B)), Dw, w ∈W ) is equivalent to Db(Rep(G)).64Appendix AAnother presentationIn this appendix, we introduce another presentation of the shifted q = 0 affine algebra, which is defined by usinggenerating series.Definition A.0.1. The shifted q = 0 affine algebra, denoted by U˙ ′0,N (Lsln), is an associative C-algebra that isgenerated by{ei,r1k, fi,r1k, ψ±i,±s±i 1k, (ψ+i,ki+1)−11k, (ψ−i,−ki)−11k|1 ≤ i ≤ n− 1, r ∈ Z, s+i ≥ ki+1, s−i ≥ ki}With the following relations (for all 1 ≤ i, j ≤ n− 1, k ∈ C(n,N), , ′ ∈ {±} and s+i ≥ ki+1, s−i ≥ ki).1k1l = δk,l1k, ei,r1k = 1k+αiei,r, fi,r1k = 1k−αifi,r, ψ+i,s+i1k = 1kψ+i,s+i, ψ+i,−s−i1k = 1kψ+i,−s−i. (A.0.1)[ψi,k(z), ψ′j,k(w)]1k = 0, (ψ+i,ki+1)±1 · (ψ+i,ki+1)∓11k = 1k = (ψ−i,−ki)±1 · (ψ−i,−ki)∓11k (A.0.2)zei,k+αi(z)ei,k(w)1k = −wei,k+αi(w)ei,k(z)1kwei,k+αi+1(z)ei+1,k(w)1k = (w − z)ei+1,k+αi(w)ei,k(z)1k(z − w)ei,k+αj (z)ej,k(w)1k = (z − w)ej,k+αi(w)ei,k(z)1k, if |i− j| ≥ 2(A.0.3)65− wfi,k−αi(z)fi,k(w)1k = zfi,k−αi(w)fi,k(z)1k(w − z)fi,k−αi+1(z)fi+1,k(w)1k = wfi+1,k−αi(w)fi,k(z)1k(z − w)fi,k−αj (z)fj,k(w)1k = (z − w)fj,k−αi(w)fi,k(z)1k, if |i− j| ≥ 2(A.0.4)zψ+i,k+αi(z)ei,k(w)1k = −wei,k(w)ψ+i,k(z)1k−wz(∑s≥0(wz)s)ψ+i,k+αi+1(z)ei+1,k(w)1k = ei+1,k(w)ψ+i,k(z)1kψ+i,k+αi−1(z)ei−1,k(w)1k = (∑s≥0(wz)s)ei−1,k(w)ψ+i,k(z)1kψ+i,k+αj (z)ej,k(w)1k = ej,k(w)ψ+i,k(z)1k, if |i− j| ≥ 2(A.0.5)zψ−i,k+αi(z)ei,k(w)1k = −wei,k(w)ψ−i,k(z)1k(∑s≥0(zw)s)ψ−i,k+αi+1(z)ei+1,k(w)1k = ei+1,k(w)ψ−i,k(z)1kψ−i,k+αi−1(z)ei−1,k(w)1k =−zw(∑s≥0(zw)s)ei−1,k(w)ψ−i,k(z)1kψ−i,k+αj (z)ej,k(w)1k = ej,k(w)ψ−i,k(z)1k, if |i− j| ≥ 2(A.0.6)− wψ+i,k−αi(z)fi,k(w)1k = zfi,k(w)ψ+i,k(z)1kψ+i,k−αi+1(z)fi+1,k(w)1k =−wz(∑s≥0(wz)s)fi+1,k(w)ψ+i,k(z)1k(∑s≥0(wz)s)ψ+i,k−αi−1(z)fi−1,k(w)1k = fi−1,k(w)ψ+i,k(z)1kψ+i,k−αj (z)fj,k(w)1k = fj,k(w)ψ+i,k(z)1k, if |i− j| ≥ 2(A.0.7)66− wψ−i,k−αi(z)fi,k(w)1k = zfi,k(w)ψ−i,k(z)1kψ−i,k−αi+1(z)fi+1,k(w)1k = (∑s≥0(zw)s)fi+1,k(w)ψ−i,k(z)1k−zw(∑s≥0(zw)s)ψ−i,k−αi−1(z)fi−1,k(w)1k = fi−1,k(w)ψ−i,k(z)1kψ−i,k−αj (z)fj,k(w)1k = fj,k(w)ψ−i,k(z)1k, if |i− j| ≥ 2(A.0.8)ei,k−αi(z)fi,k(w)1k − fi,k+αi(w)ei,k(z)1k = δijδ(zw)(ψ+i,k(z)− ψ−i,k(z))1k (A.0.9)where the generating series are defined as followsei,k(z) :=∑r∈Zei,r1kz−r, fi,k(z) :=∑r∈Zfi,r1kz−r,ψ+i,k(z) :=∑r≥ki+1ψ+i,r1kz−r, ψ−i,k(z) :=∑r≥kiψ−i,−r1kzr, δ(z) :=∑r∈Zzr.To relate this definition to the first one, we introduce the relations between generators {hi,±r}r≥01≤i≤N−1 and{ψ±i,±s±i1k} via the following(ψ+i,ki+1z−ki+1)−1ψ+i,k(z) = (1 + hi,+(z))1k,(ψ−i,−kizki)−1ψ−i,k(z) = (1 + hi,−(z))1k,where hi,±(z) =∑r>0 hi,±rz∓r.67Then we define inductivelyei,r1k :=−ψ+i ei,r−1(ψ+i )−11k if r > 0−(ψ+i )−1ei,r+1ψ+i 1k if r < −ki − 1fi,r1k :=−(ψ+i )−1fi,r−1ψ+i 1k if r > ki+1 + 1−ψ+i fi,r+1(ψ+i )−11k if r < 0ψ+i,r1k := [ei,r−ki+1−1, fi,ki+1+1]1k for r ≥ ki+1 + 1ψ−i,r1k := −[ei,r, fi,0]1k for r ≤ −ki − 1Similarly as [FT], we also predict that the two presentations in Definition 2.2.1 and Definition A.0.1 areequivalent. More precisely,Conjecture A.0.2. There is a C-algebra isomorphism U˙0,N (Lsln)→ U˙ ′0,N (Lsln) such thatei,r1k 7→ ei,r1k, fi,r1k 7→ fi,r1k, ψ+i 1k 7→ ψ+i,ki+11k, ψ−i 1k 7→ ψ−i,−ki1k,for 1 ≤ i ≤ n− 1.68Bibliography[AK1] S. Arkhipov and T. Kanstrup, Demazure descent and representations of reductive groups, Hindawi Al-gebra 2014 (2014), Article ID 823467, http://dx.doi.org/10.1155/2014/823467. → pages iii, 5, 62, 64[AK2] S. Arkhipov and T. Kanstrup, Quasi-coherent Hecke category and Demazure descent, Mosc. Math. J. 15(2015), no. 2, 257267, 404 → pages 63[AP] D. V. Alekseevsky and A. M. Perelomov, Invariant Khler-Einstein metrics on compact homogeneousspaces, Funct. Anal. Appl. 20 (3) (1986) 171–182. → pages 40[BLM] A. A. Beilinson, G. Lusztig and R. MacPherson, A geometric setting for the quantum deformation ofGLn, Duke Math. J. 61 (1990) 655-677 → pages 1[C] S. Cautis, Rigidity in higher representation theory, arXiv:1409.0827v1 [math.RT] → pages 10, 14[CK1] S. Cautis, J. Kamnitzer, Brading via geometric Lie algebra action, Compositio Math. 148 (2012) no. 2,464–506. → pages 2[CK2] S. Cautis, J. Kamnitzer, Categorical geometric symmetric Howe duality, Selecta Math. 24 (2018), no. 2,1593–1631 → pages 10[CKL2] S. Cautis, J. Kamnitzer and A. Licata, Coherent sheaves and categorical sl2 actions, Duke Math. J.154 (2010), no. 1, 135-179. → pages 2[CKL3] S. Cautis, J. Kamnitzer and A. Licata, Derived equivalences for cotangent bundles of Grassmanniansvia categorical sl2 actions, J. Reine Angew. Math. → pages 2[CL] S. Cautis, and A. Licata Vertex operators and 2-representations of quantum affine algebrasarXiv:1112.6189[math.RT] → pages 3, 1469[CR] J. Chuang, R. Rouquier Derived equivalences for symmetric groups and sl2-categorification, Ann. ofMath. (2) 167 (2008), no. 1, 245-298. → pages 1, 2, 5[FT] Michael Finkelberg, Alexander Tsymbaliuk, Multiplicative slices, relativistic Toda and shifted quantumaffine algebras, arXiv:1708.01795v1 → pages iii, 3, 6, 7, 68[H] D. Huybrechts, Fourier-Mukai transforms in Algebraic Geometry, Oxford University Press, 2006.→ pages15, 16[KL1] M. Khovanov, A. Lauda A diagrammatic approach to categorification of quantum groups I, Represent.Theory 13 (2009), 309-347. → pages 1, 5[KL2] M. Khovanov, A diagrammatic approach to categorification of quantum groups II, Trans. Amer. Math.Soc. 363 (2011), 2685-2700. → pages 1, 5[KL3] M. Khovanov, A diagrammatic approach to categorification of quantum groups III, Quantum topology1, Issue 1 (2010), 1-92. → pages 1, 5[L] M.Lubke, Stability of Einstein-Hermitian vector bundles. Manuscr. Math. 42, 245-257 (1983) → pages 40[Lu] G. Lusztig, Introduction to quantum groups. Birkhauser, Boston, 1993. → pages 8[OSS] C.Okonek, M.Schneider, and H.Spindler, Vector bundles on complex projectivespaces(Birkhauser,Boston, 1980). → pages 40[R] R. Rouquier, 2-Kac-Moody algebras, math.RT/0812.5023. → pages 170

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