Towards a Functor Between Affine and Finite Hecke Categories in Type A by Kostiantyn Tolmachov

Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy

at the

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June 2018

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Towards a Functor Between Affine and Finite Hecke Categories in Type A by Kostiantyn Tolmachov

Submitted to the Department of Mathematics on May 1, 2018, in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Abstract In this thesis we construct a functor from the perfect subcategory of the coherent version of the affine Hecke category in type A to the finite constructible Hecke ca- tegory, partly categorifying a certain natural homomorphism of the corresponding Hecke algebras. This homomorphism sends generators of the Bernstein's commutative subalgebra inside the affine Hecke algebra to Jucys-Murphy elements in the finite Hecke algebra. Construction employs the general strategy devised by Bezrukavnikov to prove the equivalence of coherent and constructible variants of the affine Hecke category. Namely, we identify an action of the category Rep(GL,) on the finite Hecke category, and lift this action to a functor from the perfect derived category of the Steinberg variety, by equipping it with various additional data.

Thesis Supervisor: Roman Bezrukavnikov Title: Professor of Mathematics

3 4 Acknowledgments

I would like to thank my advisor, Roman Bezrukavnikov, for his support and guidance over these years, as well as for many helpful discussions and sharing his expertise. This thesis would be impossible without his help on so many levels. For the most part, it can be considered an etude on the methods of his paper lBez16]. A beautiful idea to use Lemma 4.2 for the proof of the main theorem is also due to him. I am very glad to have this opportunity to thank Michael Finkelberg for his constant support and encouragement during the darker times of my graduate studies and beyond them. I am grateful to Ivan Loseu for his support and for sharing his life wisdom in a critical moment. I would also like to thank Pavel Etingof, Michael Finkelberg and Ivan Mirkovi6 for helpful discussions. I would like to thank Cristina Iscenco for her help in editing the manuscript. Last but not least, I wish to express gratitude to Cristina and all my family for their caring presence and patience. Dedicated to my grandmother, Elena.

5 6 Contents

1 Introduction 11 1.1 A lgebraic setting ...... 11 1.2 C ategorifications ...... 12 1.3 Jucys-Murphy sheaves ...... 13 1.4 Plan of the categorification of the central arrow ...... 14 1.5 Lie algebra version ...... 15 1.6 The functor from the perfect derived category of St ...... 15 1.7 Relation to knot invariants, Hilbert schemes and matrix factorizations 16 1.8 Organization of the paper ...... 16

2 Algebraic preliminaries 19 2.1 H ecke algebras ...... 19 2.2 Elementary symmetric polynomials in Jucys-Murphy elements . . 21 2.3 Anti-spherical projector ...... 21 2.4 Universal property of Rep(GL) ...... 21

3 Geometric setup 23 3.1 Sheaves on the base affine space ...... 23 3.2 Harish-Chandra functor ...... 24 3.3 Parabolic restriction and induction ...... 27 3.4 Constructible finite Hecke category and character sheaves ...... 27 3.5 Braided structure and the twist ...... 29 3.6 Sheaves on tori ...... 29 3.7 Affine Hecke categories ...... 30

4 Construction of the functor from Rep(GL,,) 33 4.1 Sheaves on the unipotent variety ...... 33 4.2 M ain theorem ...... 33 4.3 Lie algebra version of the main theorem ...... 38 4.4 Another description of the isomorphism in the main theorem . .... 39

5 Extension to the functor from the perfect derived category 41 5.1 Lowest weight arrow and monodromy endomorphism ...... 43 5.2 Filtration by Jucys-Murphy sheaves ...... 45 5.3 Passage from Cst to St ...... 45

7 5.4 Compatibility with the monoidal structure ...... 46

A Remark on the Kirillov-Pak character formula 47

8 List of Figures

1-1 Homomorphism on the level of braid groups ...... 11

9 10 Chapter 1

Introduction

1.1 Algebraic setting

This thesis is concerned with the categorification of a certain homomorphism between the affine and finite Hecke algebras in type A, in the setting of known geometric categorifications of the algebras themselves. The homomorphism in question may be most easily seen on the level of the cor- responding braid groups. If we interpret the braid group B, on n strands as the fundamental group of the configuration space Confn(C) of n points in the complex plane C, and the affine braid group Baf as the fundamental group of the configura- tion space Confn(C*) of n points in the punctured complex plane C*, based at some configuration ( not meeting 0 E C, then the inclusion C* -+ C gives the homomorpism

Bnf ~ ri(Confn(C*), () - ir1 (Confn(C), () B, see Figure 1-1. This descends to the homomorphism

between the corresponding extended affine Hecke algebra of the reductive group GLn and the finite Hecke algebra. Recall that iKJa contains Hn as a subalgebra. It also

:9

Figure 1-1: Homomorphism on the level of braid groups

11 contains a large commutative subalgebra, defined by Bernstein, which is identified with the group algebra of the (co-)weight lattice of GL,. The homomorphism H has the following remarkable property: under some choice {ei, ... , e} of generators in this lattice, the corresponding Bernstein generators {1, ... , 0,} map to the multiplicative Jucys-Murphy elements {L 1 ,... , L, in the finite Hecke algebra. 11 is identity when restricted to H, c Nt". Symmetric polynomials in 011 are known to span the center 3(HIf) of Nta, see [Lus83], and so are symmetric polynomials in Li for HJ, see [DJ87]. We get the following diagram

(1.1)ta n3 j

3(Hn) >Hn All corners of this diagram have their geometric categorifications.

1.2 Categorifications

In what follows we work over the field of the complex numbers C. All the categories considered are C-linear and all the varieties are defined over C. Categorification of the finite Hecke algebra comes in many forms, of which we will use some completed variant of the monodromic category DJ of constructible sheaves on the base affine space for GLn. The categorification of the center is the completed variant of the derived category of unipotent character sheaves on GLn, Db(CS): in fact, it is known that the appropriate variant of the category of character sheaves is the Drinfeld center of the finite Hecke category. See [BFO12] for the statement in the setting of abelian categories of D-modules, [BZN09 for the statement in the derived setting, and [Lus15] for the related statement in arbitrary characteristic. The inclusion of the center is given by the so-called Harish-Chandra functor, which we denote ljc, introduced by Lusztig. See also [Gin89I, where the relation to central objects in the Hecke category is noted. While categorifications of the finite Hecke category all come from the world of con- structible sheaves (or D-modules), one can obtain a categorification of the extended affine Hecke algebra both in constructible and coherent settings. Namely, our con- structible avatar of the affine Hecke category is the completed monodromic variant of the derived category of constructible sheaves on the affine flag variety of GLn, equiv- ariant with respect to the (radical of) Iwahori subgroup, which we will denote simply by D'(Flaff) in this introduction. It is a result of [Bez16j, that the constructible affine Hecke category is equivalent to (the completed variant of) the derived category of the equivariant coherent sheaves on the Steinberg variety, which we will denote by Db(CohG(St)) in this introduction. Under this equivalence, the structure sheaf Og corresponds to the pro-unipotent tilting sheaf of the open orbit, to. Categorification of the center of the affine Hecke algebra is known as a Satake category PG(O) (Gr) of equivariant perverse sheaves on the affine Grassmannian cor-

12 responding to GL,, see [Lus83I, [MV07]. It was shown in op.cit., that this category is equivalent to the category Rep(GL,) of representations of GL,. The corresponding inclusion was categorified in [Gai0l] (note that Rep(GL,) is much smaller than the actual Drinfeld center of the affine Hecke category). All the categories discussed are equipped with the monoidal structure, given by convolution, which we denote by * in this introduction. We obtain the following diagram:

[MV07 [Gai0l] b [Bez16 Rep(GL,) > 'PG(0)(Gr) > D,(Fl') [ Db(CohG(St)) (1.2)

- Db(CS) bf k<

Conjecture. There are monoidal functors replacing dashed arrows in the diagram (1.2), making it commutative and categorifying the diagram (1.1). In this thesis, we prove the following weakened form of this conjecture. We have a perfect variant DGef(St) of the coherent Hecke category. Since the Steinberg variety is singular, it is not equivalent to the full derived category. Our main result is the following Theorem (see Theorems 1, 3). There are monoidal functors =z, =, making the following diagram commutative:

Rep( GLn) i a0 4() Dpef( st) ,zi (1.3)

Db(CS) ()To-+D

It is a categorification of a part of the diagram (1.1). In the rest of the introduction, we describe the plan of the proof of this theorem.

1.3 Jucys-Murphy sheaves

The fact that IH is defined already for braid groups allows us to identify the desired images of the so-called Wakimoto sheaves categorifying Bernstein's generators. We have a standard basis {t} of the finite Hecke algebra, indexed by elements w of the Weyl group. A choice of a set of simple reflections {si} gives a set of generators ti := t,2 . One can pick the generators {0} of Bernstein's commutative subalgebra so that I1(0 1) =te 1 =:L1,

fl(Ok+1) = tkLktk =: Lk+1-

Elements Lk are well-known in the study of Hecke algebras, and are called the multiplicative Jucys-Murphy elements. See [10051 for a review.

13 Basis {tj has a natural categorical analogue in the Hecke category bf, given by the so-called standard sheaves A, (for the precise definition and notations see the main body of the paper). We can readily define

L := Ae,

Lk+1 := Ak * Lk * Ak. We call the complexes Lk Jucys-Murphy sheaves. Remark. Let wo be the longest element of the Weyl group of GL,. Convolution with the complex Ae * A O defines the inverse of the Serre functor S, on the Hecke category, see IBBM04I. Let w' be the longest element of the Weyl group of GL,_ 1 x Gm embedded into GL, as a block-diagonal subgroup. We have

2 I Ln *.F = A*2 A*(- ) * F = S- S_ 1 (), so nth Jucys-Murphy sheaf can be interpreted as a "ratio" of two Serre functors. Compare IBK90, Proposition 3.9. We don't make any use of this fact below.

1.4 Plan of the categorification of the central arrow

The main contribution of this thesis is Theorem 1, which may be of independent interest. In it, we construct a functor from Rep(GL.) to the completed derived category of unipotent character sheaves. We use the following universal property of the category Rep(GLn): to give a monoidal functor from Rep(GLa) to a symmetric monoidal pseudo-abelian category C, it is enough to specify an object V of C, satisfying A"+1 V = 0 and such that AnV is invertible. It will be an object corresponding to the standard representation V of GLn = GL(Vn), from which one gets other representations by applying Schur functors and tensoring with powers of A"K. See Section 2.4. We will apply this reasoning to the derived category of adjoint-equivariant con- structible sheaves on GLn. This category is monoidal, product being given by convo- lution, and can be equipped with a symmetric braided structure, see Section 3.5. In our algebraic picture, object V should correspond to the element

Ui -_01= Li.

We would expect, then, our object V in the category of character sheaves to have a filtration by Jucys-Murphy sheaves, after we take it to the Hecke category. This object turns out to be the parabolic Springer sheaf, see Chapter 3.2 for definition, or, more precisely, its projection to the category of character sheaves. See also a related computation in the Appendix. Recall that GLn acts on the variety jV of its unipotent elements by conjugation, and orbits of this action are numbered by partitions of n, according to the Jordan

14 decomposition. Let ICA be the intersection cohomology sheaf of the orbit numbered by a partition A. Let Ak be a "hook" partition (k, 1,... ,1). We set A, = (n). Let Sprp ~_IC,\ e ICA. We have the following Theorem (Section 4.2). a) AkSprp ~ (ICAk e ICAk+l), for 0 < k < n.

b) AnSprP ~ ICn

c) An+lSprp = 0. Here the exterior powers are with respect to the monoidal structure given by the convolution on GL,. The derived category of unipotent character sheaves is closed under convolution, but lacks a unit object. It is for this reason we must pass to the completed cate- gory, where the unit object 6 exists. Convolution with 6 gives a projection from the full equivariant derived category D' (GLn) of constructible sheaves on GLn to the completed category of unipotent character sheaves. In Section 4.2 we will prove that AnSprP * 6 is an invertible object in the latter category, and so construct a functor from Rep(GLn) to it via the strategy indicated above.

1.5 Lie algebra version

Although it is not directly related to the categorification of the homomorphism in question, we also prove a variant of the theorem of the previous section in the setting of equivariant sheaves on the Lie algebra of GLn. Here the proof, employing the notion of the Fourier transform of sheaves on vector spaces, is similar, but more direct. Namely, identify N withe the nilpotent cone A' C gf, = Lie(GLn), and again denote by ICA the intersection cohomology sheaves on the corresponding orbits. Let

Stp 'CA IC.A 2 We have the following

Theorem (Section 4.3). a) Ak'sprp ~ (iCA,( e ICAk+l), for 0 < k < n.

b) Ansprp ~ IC c) An+1sprp = 0. Here the exterior powers are with respect to the monoidal structure given by the additive convolution on gfn.

1.6 The functor from the perfect derived category of St

We are in the following situation: we have a monoidal functor from the Satake cate- gory Rep(GLn) to bf, as well as (products of) Jucys-Murphy sheaves there, categori- fying the image of the Bernstein's lattice of the affine Hecke algebra. Constructions

15 of [Gai0l] and [AB09] predict that we should have two additional pieces of data: monodromy endomorphism of objects coming from the Satake category, and a com- patible filtration of these by Jucys-Murphy sheaves, corresponding to the filtration of Gaitsgory's sheaves by Wakimoto sheaves. Presence of this additional data, which we construct in Section 5 from the fact that our central sheaves can be obtained by some parabolic induction and restriction procedure, puts us in the situation of [Bez16. In op.cit., such data was used to construct a functor from the coherent version of the affine Hecke category to its constructible version. We will employ it to construct a functor from (the perfect part of) the coherent version of the affine Hecke category to the finite Hecke category Df.

1.7 Relation to knot invariants, Hilbert schemes and matrix factorizations

There is a body of work on various geometric constructions of knot invariants, which seem to be related to the categorification of the homomorphism H. One of the initial motivations of this thesis project was to understand some struc- tures arising in the recent work [GNR16]. In particular, there is the following con- jecture. Let FHilbdg be the flag Hilbert dg-scheme of [GNR16], and let SBim" be the category of Soergel bimodules, see [Soe07]. Homotopy category Kb(SBimn) (or, rather, its completed variant) is another categorification of the finite Hecke algebra, equivalent to the one we use, see [BY13]. Conjecture ([GNR16]). There is a pair of adjoint functors

K'(SBim,) *,iD' (Cohc* xC* (FHilbnd(C)))

t* being monoidal and fully faithful. In particular, functor from the conjecture above is predicted to send certain line bundles on FHilbd9 to similarly defined Jucys-Murphy objects in Kb(SBimn), just as in our conjecture similar line bundles on the diagonal in St are predicted to be sent to Jucys-Murphy sheaves in Df. In the work [OR171, a functor between two convolution algebras of equivariant matrix factorization is constructed, and is shown to intertwine the braid version of the homomorphism H for the images of affine and finite braid groups inside the corresponding convolution algebras. This is also applied to the computation of knot homology. At this point we do not know any direct relation of the functors constructed in this thesis to the above settings.

1.8 Organization of the paper

This thesis is organized as follows.

16 In Chapter 2 we describe the algebraic setting which we would like to categorify and give some supporting computations. In Chapter 3 we define the geometric setup with which we will be working, both in constructible and coherent settings, and recall the necessary facts and constructions involving character sheaves. In Chapter 4 we construct a functor between central categories, and discuss a variant of the main theorem for Lie algebras. In Chapter 5 we extend this functor to a functor from the perfect derived category. In the appendix, we make a remark about a relation of multiplicative Jucys- Murphy elements to the Kirillov-Pak character formula.

17 18 Chapter 2

Algebraic preliminaries

2.1 Hecke algebras

Let W = S, be a finite Weyl group of type A, identified with the symmetric group on n elements, let I C W be the set of simple reflections,

I =- {(1 2), (2 3), ... , (n - I n)}.

Let / : W -* Z+ be the length function. The finite Hecke algebra H(W) = H, is a unital algebra over Z[v, v 1 ] generated by elements t5 , s E I, subject to the following relations. Denote ti = t(i(i+1)).

I. titi+1ti = ti+1titi+1.

2. titj = tjti,\It - J| > 1.

3. t= 1 + (v 1 - v)ti.

H has a basis tw, w C TV, defined by t = t, ... ts, for a reduced expression w =

S 1 . .. Sk, and the Kazhdan-Lusztig basis C., w E W, see [KL79}. Let (X*,

X* = X* =: X ~ Z' = spanz{ei,-, , enI}, =V = {ei - e3 } is c X. W acts on X and D permuting ei. Fix the set of simple roots A= {ei - ei_+1}n . This defines the set of dominant weights to be X+ ={(A-,... , An), Ak A-k+1 for all k}. The extended affine Hecke algebra IH1a, corresponding to this root datum, is a unital algebra over Z[v, v-1] generated by elements t8, s E I, 0., x E X, subject to the 0 following relations. Denote ti = t(i(i+l)), Oi = er

l. titi+1ti = tj+1tjti+1-

2. titj = tjti ,ji -j > 1.

1 3. t2 = 1 + (v - v)ti.

19 4. OxOy = Ox+y.

5. tj Oj = Oj tj if j 7 i 7i + 1.

6. ti0iti = Oi+1.

7. 0o = 1.

For an element A E X, let WA be its W-orbit. Following [Lus83I, define

ZA 0'O. A'EWA For a dominant A E X and p < A, define d(p; A) to be the multiplicity of the weight ,a in the finite-dimensional irreducible representation VA of GL, with the highest weight A. As in op.cit., define s = d(p; A)zA.

We have SvSv; m(A, A', A")svy', A"t where m(A, A', A") satisfies

V ThemVAA' 0EVm(A,AA") =deto J. Ber

We have the following Theorem due to J. Bernstein:

Theorem ([Lus83I). The center 3(kiIf) of the extended affine Hecke algebra is a free Z[v, v-1 ]-module with a basis ZA, A C X+.

Let H1: k'~ -+ H, be the homomorphism defined by

11(ts) = tS, s E17

H(01) = 1. It is easy to see, that 1I is indeed a homomorphism, is defined uniquely, and that

rl(Ok) = tk-ltk-2 t2tit2...tk-2tk-1.

We denote Li := 11(02) and call these elements Jucys-Murphy elements in the finite Hecke algebra. We have the following theorem:

Theorem ([DJ871,[FG06|). Symmetric polynomials in elements Li lie in the center 3(H,) of the finite Hecke algebra. Moreover, the restriction of the homomorphism I : 3(Hnff) -+ 3(Hn) is surjective.

20 2.2 Elementary symmetric polynomials in Jucys-Murphy elements

The following interpretation of the elementary symmetric polynomials ek in Jucys- Murphy elements will be useful to us. Let V be the standard n-dimensional repre- sentation of GL, GL(V). We have

H(sAykV) = ( S O) = ek(L, ... , Ln). A'EW(e1+---+ek)

Let W' Sk x Sn-k C S, considered as a parabolic subgroup with simple re- flections Si, . .. , s Sk+1,... ,s n-1, and let Wk be the set of minimal length rep- resentatives of W/W'. Let WO,k be the longest element in Sk (with respect to the choice {s,... , sk_1} of simple reflections). Note that L1 ... Lk twok. We record the following straightforward computation: Lemma 2.2.1. ek(L1,...,Ln) Z Wt,kt W1 wEW'

2.3 Anti-spherical projector

Let wo be the longest element of W,. Consider an element

S=V-1(w0) E (-V )1W)t W. w EW We have Cw = 0, unless w = 1. Let j~laff - 1 aff - taff n,perf - It~' S Hn Let Hperf be the restriction of H to IHeaff We have Hper(ab) (ab) ca, n,perf* ef<) 71a 1 where Cab E [v, v- ] is a coefficient of C1 in ab. Our main result can be considered a categorification of the projection 1lper.

2.4 Universal property of Rep(GLn) Let C be a C-linear, symmetric monoidal pseudo-abelian category. Then, for any object X E C and positive integer k, we have the Sk-action on the object Xk, and, for any partition A, Schur functors SAX are defined as images of the corresponding projectors.

Lemma 2.4.1. Let X E Ob(C) be such that A'X is invertible and An+ 1X = 0. Then there is a monoidal functor Rep(GLn) -+ C

21 sending the standard n-dimensional representation of Rep(GL,) to X.

Proof. The fact that AnX is invertible implies that X is a rigid object of dimen- sion n. Thus, by the result of [Del07], there is a functor from the Deligne category Rep(GL(t = n)) to C, sending the generating object to X. The fact that An+ 1X = 0 implies, by the second fundamental theorem of invariant theory of the general linear group, that this functor factors through Rep(GLn). E

22 Chapter 3

Geometric setup

3.1 Sheaves on the base affine space

Contents of this section is standard, see e.g. [Lus15], [BY13J. If X is an algebraic variety with an action of an algebraic group G, let Dn (X) be a bounded constructible G-equivariant derived category of constructible sheaves on X. If G is trivial, we write simply Db(X). We write CX for the constant sheaf of rank 1 on X. Let G be a connected reductive group over C, B its Borel subgroup, T C B the maximal torus in it, U its unipotent radical. Let B= G/B, let Y = G/U. We have a right T x T-action on Y x Y:

(ti, t 2)(xU, yU) = (Xt1U, yt2 U). Let Y X Y Y= T ' where T acts on Y x Y diagonally on the right. More generally, if P C G is a parabolic subgroup, Up its unipotent radical, L C P a Levi subgroup, let Bp = G/P, Yp = G/Up, G/Up x G/Up YP = L G acts on Yp diagonally on the left. Let W = N(T)/T be the Weyl group of G, let I C W be a set of simple reflections, let 1 : W -+ Z+ be the length function. We have the natural projection 7r : Y x Y -+ B x B, which factors through Y. Let 0, be a G-orbit in B x B corresponding to w E W, and let 6 = 7r-(OW). Let OI Ow be the corresponding closures, and jh : Ow -+ Y x Y the corresponding locally-closed embedding. Let gw be a lift of w E W to N(T). We have the projection pgw : -* T, defined

23 by the inclusion 1 - y E Ug-p 9 ,(xU, yU)U. For a local system L on T, let

AL = Jw pL[dimT + dimY + l(w)],Vi =3.pL[dimT + dimY + l(w)].

A, V7 do not depend on a lift gw (up to a non-canonical isomorphism). We will denote simply by AL, VL members of their isomorphism classes. Let j* : pg' (1) - Y x Y be the obvious locally-closed embedding. Denote

Jw 1 (1)[dima Y + I(w)].

Let pij : Y x Y x Y -+ Y x Y be a projection on i'th and j'th factors. For T,g E D'(Y x Y), let

T * g = P1,3! (P*,,T 0 p* 3!) [- dim Y].

This equips Db(Y x Y) with the monoidal structure. Convolution Yp x Yp is defined in the same way. Db(yp), considered as the subcategory of L-equivariant objects, inherits the monoidal structure. We record the following simple

Lemma 3.1.1. Let L be a W -invariant local system on T. Then Af * A*, A*, A' ~.A'. Here e E W is the unit element.

3.2 Harish-Chandra functor

Following [Lus851, consider the following diagram:

G x Yp f (3.1) G YPxYP

with 7G(g, XP) = g, f(g, XUp) = (xUP, gxUp), and its version obtained by taking the quotient with respect to the free right L-action:

G x Bp 7rGf (3.2) G Y

with 1G(g, XP) = g, f(g, xP) = (xUp, gxUp). We have functors

= f [dimY],ip =G! f dimYp,

24 and Scp = f!Tr [dim Bp],xp =rG!f[-_ dim BP]. When P = B, we simply write

jCp = )C, XP = x.

We will also need intermediate functors, see [Lus04]. Let P c Q be two parabolic subgroups.

PQ (3 3) YQXYQ YpXYP Here

ZP,Q = {(g1UQ, g 2UQ, XUp) : g1-1X C Q1, 1 WQ(g1UQ, g2UQ, XUP) = (1UQ, g2 U), f(g1UQ, g2UQ, XUp) = (XUp, g2g xUp). Let WrQ, f be the maps obtained after dividing WF, f by the free right LQ, Lp actions, respectively. Define

CP = f!7rQ[dimYp - dim YQ], X, = WrQ f* [dimYQ - dim Yp],

and f!rX[dimB - dim Bf, =rQ f*[dim BQ - dim Bp]. By the result of op.cit., for P D Q D R, we have OcQ o OCP =cP and that Ocp(F) = 0 implies T = 0.

: x G -+G be the multiplicationLet morphism,G let iri, r2 G x G -+G be the projections on the corresponding factors. For F, 9 c Db(G), let F * g = (7r*F 7r*g). Let AVP = {(g, XP) : g E xUpx-1}

be the parabolic Springer resolution, and let 7r : Hp - G be the natural projection. Define Sprp = C[2 dim Up] - the parabolic Springer sheaf. Recall from [Gin89J, that cp, Fcp are monoidal functors, and that we have Xp 0 (cp(F) = X o cp(F) =F* Sprp. In particular, identity functor is a summand of xP 0 JCp- Fix J C I and let Wj c W be the corresponding parabolic subgroup. Assume that P is conjugate to the parabolic subgroup Pj corresponding to J. Write WJ for the set of minimal length representatives of W/Wj. For every w E WJ, choose a lift gw to N(T).

For an objects X, {C} of a triangulated category, we write X E (C1, C2 , .-- , Ck) if there exists a sequence of objects {Xi}f_ 1 , X1 = X, Xk = Ck, such that for all

25 i < k, (Ci, Xi, Xi. 1) is a distinguished triangle. The following lemma is a straightfor- ward consequence of standard distinguished triangles for 6 functors for constructible sheaves.

Lemma 3.2.1. Let X be a variety stratified by locally closed subvarieties {St}U_ 1, and let Jt : St - X be the corresponding locally-closed embeddings. Assume that Sk C Si implies k > 1. Then for every F E Db(X), F E (jt!j7)" .

The following lemma will be used to construct a filtration of certain central sheaves by Jucys-Murphy sheaves:

Lemma 3.2.2 (cf. [Gro92, [Lusl3l). (c(Sprp) E (CA* A w )Gwj, where objects are in some non-increasing order with respect to the Bruhat order on {w}.

Proof. Let X = Rp x Y, and write

q$: X - Y x Y, (g, xP, yU) = (yU, gyU).

By base change, (c(Sprp) !Cx[2dim Up]. Applying Lemma 3.2.1 to the filtration by

X" ={(g, xP, yU) C X, x-y E PwB},

for w E W-, we get that

c(Sprp) G (qO!C[2dimUp])wswj,

where 0, is a restriction 0 to Xw. We now claim that Xw is a bundle over the variety

C9W= {(xU, yU, zU) : (xU, yU) E 6w,(yU, zU) e 6w , pg9 (xU, y U) = PgW (YU, zU) = with a fiber Adim-UP(w). Namely, the fibration map 7rxw sends (g, xP, yU) E X" to (yU, zU, gyU), where zU is a unique element of Y such that

(yU, zU) E 6w, pg,(yU, zU) = 1.

To check that this defines a map Xw -4 C,, we must verify that pg- 1(zU, gyU) = 1. Indeed, we have z 1 gy E z- 1yUpy-yU = z 1 yU. Since pg (yU, zU) = 1, we have y-1 z E UgWU, so z 1 y E UgW 1U, and we are done.

Note that 7ri,3!_CcW =Ag * Ag, where 7ri, 3 ! is the restriction of the projection 2 y3 _ y to CW. We also have q = r,3 0 7rX. So ow!C!xw[2 dim Up] = Ag * A

26 3.3 Parabolic restriction and induction

Let P C G be a parabolic subgroup, L be its Levi quotient. Let

Op = {(g,xP) E G x 13p: g E xPx- 1}

be the parabolic Grothendieck-Springer variety. Consider the standard map 7rL :p -+ L, and a map f : Op -+ Yp given by the restriction of the map f from (4.1) from G x Bp to CF. We have the monoidal functor iL Db(L) -4 Db(Yp), iL = f!wr[dim Bp]. In fact, it gives an equivalence from Db(L) to the subcategory of complexes in Db(yp) supported on the closed subvariety Dp = {(xUp, yUp)L : xP =y} C Yp. Let i" be 6 the inverse equivalence. Let p :D -+ Yp be the corresponding closed embedding. We have the following description of the parabolic restriction and induction functors, first defined in [Lus85]:

ResL(F =L P 7olc()IndLg =XPZL (9)-

We will also use another description of induction and restriction functors, also due to Lusztig. See also [Gin93]. We have that Gp is a G-equivariant fiber bundle on G/P with a fiber P. The restriction to the fiber gives a bijective correspondence between G-equivariant com- plexes on Gp and ad-P-equivariant complexes on P. If we equip Db(op) with a !-convolution structure along G, it is easy to see that the above correspondence be- comes monoidal. For an object A E Db (P) write A for the corresponding object in DG(Cp). Let tp : P -+ G be the corresponding closed embedding, let qL : P -+ L, -r : Op G be the natural projections. We have

Res G A= qL!t* A = qL t A,

IndG A= -rFLA[2 dim Up].

3.4 Constructible finite Hecke category and charac- ter sheaves

Y x Y is a T x T-torsor over B x B. Thus it makes sense to talk about T x T- monodromic sheaves on Y x Y. For a coset A + X of a weight A of Lie T, let PA be the category of G-equivariant T x T-monodromic perverse sheaves on Y x Y with generalized monodromy A x -A. If A = 0 we will call this category unipotent and denote simply by P. Let Df = DbP. Our version of the Hecke category is the free-monodromic constructible derived category Df := D (T x T\Y x Y)

27 defined in [BY13]. It has a perverse t-structure of its own, with the heart 'P D P, and one has bf = Db(). Remark. Note that both Df and bf are monoidal categories. However, Df has no unit, while bf has one, see below. It is for this reason we choose to work with Df. Our results could also be reformulated for certain finite quotients of pro-unipotent sheaves involved, bearing the just mentioned defect of the finite category in mind. bD has the subcategory T of free-monodromic tilting objects. Convolution with such an object is an exact functor. It is shown in op.cit. that

bf ~ Ho(T).

There is a collection T, w E W, of indecomposable free-monodromic tilting sheaves. Moreover, 'Two satisfies the following property:

The category of unipotent character sheaves, CS, is the category of G-equivariant perverse sheaves F on G such that (jc(.T) E Dbp. By the result of [MV88], this definition is equivalent to the standard one. Similarly, we can consider completed categories CS, bG - Db(CS). Let L1 be the free pro-unipotent local system on T, see [BY13I. A sheaf Ai" on Y is the monoidal unit in the category of Db(y). It is the image under rc of the perverse sheaf S E CS. S is the monoidal (pro)-unit in the category of unipotent character sheaves on G. We have the following

Lemma 3.4.1 ([BY13I).

a) AS'* V~AfA,, V W/1*Vi~ VWW, if I(w) + 1(w') l(ww'). b) At' * Vti Vtl * At ~At,.

c) DAi ~ Ve'.

In the same way, if P D B is a parabolic subgroup with a Levi subgroup L, and A is a weight fixed by W(L), we have the pro-local system on T of "infinite Jordan type" A, 4, see [Gin88J, and a corresponding sheaf AtA. It is an image under jcp of the parabolic character sheaf 6,\ on Yp. When we say that some object X is invertible with respect to the convolution in the category Df (or in the category of unipotent character sheaves), we mean that there is an object Y in the corresponding category, such that

(X * Y ~_Y * X ~ 6 respectively).

28 3.5 Braided structure and the twist

Here we recall the necessary constructions from [BD14J. We will need a notion of the braided structure on the G-equivariant derived cat- egory of constructible sheaves on G. Define : G x G - G x G, (g, h) --* (g, glhg),r : G x G -+ G x G, (g, h) - (h, g). We have p o = p o r. For A, B E D'(G), we have A Z B ~ (! (A Z B) by G-equivariance of B. So

A * B = p!(A ZB)~p (A Z B)=- p! (A Z B)~- B* A.

This defines a braided structure on D' (G), denoted by

A,: A * B 4 B * A.

For any M E D'(G) we have an isomorphism 0 M : M - AM called a twist, see op.cit. It satisfies OA*B = !A,B 0 fB,A o (OA * OB)- On the stalk of M at a point g E G it acts by the conjugation by g E ZG(g) - centralizer of g in G. Corollary 3.5.1. Assume that centralizers of all points in G are connected. Then D'(G) is symmetric, meaning that

3 3A,B 0 B,A ~ IdB*A for all A, B E D'(G).

3.6 Sheaves on tori

We recall several facts about the convolution of constructible sheaves on tori, see [GL96j. For a weight A of Lie T let C) be the corresponding local system on T.

Lemma 3.6.1. Let [' : G, -+ T be a coweight of T such that (A, pv) g Z. Then, for any constructible sheaf F on T, equivariant with respect to the Gm-action defined by y', we have 4) * F = 0. Proof. It is enough to consider the case T = Gm, in which case the statement is equivalent to the fact H*(C) = 0 for A $ 0 E C/27riZ.

Lemma 3.6.2. Let F E Db(T) be such that F * x = 0 for all A # 0. Then for all i, perverse cohomology sheaves PH2F are shifted local systems on T with unipotent monodromy. Our proof is a direct adaptation of the proof of Proposition 3.4.5 [GL961. We state this proposition and another lemma from op.cit., originally due to Laumon.

Lemma 3.6.3. Let F E Db(T) on T such that H*(F 9 4)) = 0 for all A. Then F =0.

29 Proof. This is Proposition 3.4.5 in op.cit.

Lemma 3.6.4. Let F E Db(T) be perverse. Then

X(T) = E(-1)'Hc(.F) ;> 0. i

Proof. This is Corollary 3.4.4 in op.cit.

Proof of Lemma 3.6.2. First note that the condition F * LA = 0 is equivalent to the condition H*(F 0 LA) = 0. Assume that the statement is known for tori of rank < n, and let n = rk T. Let T = T' x T", where T', T" have smaller ranks. Then, for any weight (A', A") of T' x T", projection formula gives

rTI(F 0 Lcv\,A11)) -_ 7T~T(F 09 L(o,,A")) 09 LA/.

So, if A" # 0, we get, by Lemma 3.6.3, that rT' (TF9L(o,Au)) = 0, so that F is smooth with unipotent monodromy along all fibers of lrFT. Since T' was chosen arbitrarily, this implies that F is smooth with unipotent monodromy. We now need to prove that the result holds for Gm. Consider the lowest non-zero perverse cohomology sheaf A = PHmn (F). Let A' c A be an irreducible subsheaf. Since H,(.) is t-exact from the left, we get that H,(A' 0 LA) = 0. This means that A' can not be a punctual sheaf, so it is a shift of a (non-derived) *-extension of some local system on a Zariski open subset U c Gm. We also have X(A' 0 LX) = HO(A' 0 Lx) - Hcl(A' 0 LA) < 0. By Lemma 3.6.4, this implies that X(A') = 0, so, by the Euler-Poincare formula, A' is either 0 or a shifted rank one local system on Gm. Since H2(A' 0 LA) = 0 for A # 0, this implies that A' is CG [1]. Then H (A/A' 0 LA) = 0, and we can repeat the argument by induction on length of A to get that A is a shifted unipotent local system. Again, this implies that H*(Tmn+lF 0 LA) 0, and we're done by induction.

Corollary 3.6.1. Assume that for F E D'(G) we have c(F) * Aft)' - 0 for every A # 0. Then Fc Db(CS).

3.7 Affine Hecke categories

Here we explain the relations between various variants of the affine Hecke category. Let AC= C((t)) D 0 = C[[t]]. Let I be an Iwahori subgroup of GLn(IC), let IP be its pro-unipotent radical. Let Flaff = GLn(K)/I be the affine flag variety.

Let 1 = GLn(K)/I0 . We have I/I' - T and Fl is a T-torsor over Flaff with respect to the right T-action.

30 0 - aff Let Do0 o be the full subcategory of 1 -equivariant sheaves on Fl consisting of complexes whose cohomology is monodromic with respect to the above T-torsor structure. Let b be the pro-completion of the category Djojo defined in [BY13j. Both categories D and Djolo carry a monoidal structure. Remark of Section 3.4 applies here equally. Recall from [Bez16] that we have the spectral variant of the affine Hecke category. For an algebraic variety X, let Coh(X) be the category of coherent sheaves on X. For a closed subscheme Z c X, let Cohz(X) be the full subcategory of Coh(X) consisting of sheaves set-theoretically supported on Z. If X is equipped with an action of an algebraic group G, let CohG(X) be the category of G-equivariant coherent sheaves on X, and let DSr(X) C Db(CohG(X)) be the subcategory of perfect complexes. Let g be the Lie algebra of the Langlands dual group G', let g be the corresponding Grothendieck-Springer resolution, St = 0 x . Let K c g- be the nilpotent cone. Let g be the spectrum of the completion of the ring of functions O(g) at M. We have a completed variant of St, St = St x,_g. Categories Db(Coh.V(St)), Db(CohG(St)) have a natural monoidal structure given by the convolution. We have the following

Theorem ([Bezl6]). The following are equivalent as monoidal categories:

Db(CohG(St)) ~ Dolo ,

Db(CohG(St)) b.

It is explained in op.cit. that the category D r(St) can be thought of as a categorification of the perfect subalgebra Iiff see Section 2.3.

We have two natural G -equivariant projections r, 2 : St -+ B, and, for any two weights A,, A2 of T and corresponding equivariant line bundles O(Qi), O(A 2 ) on B, we have a line bundle 0-t(A,, A2 ) = 7r*0(A) ost 7rO(A2 ) c DG(St). For a representation V of G-, we have a corresponding G-equivariant bundle V & Ot.

31 32 Chapter 4

Construction of the functor from Rep(GL,)

4.1 Sheaves on the unipotent variety

In this section G = GL,. Let V be the standard n-dimensional representation of GL,= GL(V,). Let P be a parabolic subgroup of GL, fixing a vector in P(V.). Recall that G acts on the variety MA, of unipotent elements in G by conjugation, and orbits of this action are numbered by partitions of n, according to the Jordan decomposition. Let ICA be the intersection cohomology sheaf of the orbit numbered by a partition A. Let Ak be a "hook" partition (k, 1,... , 1), and let A, = (n). Then Sprp ~_IC) 1 E ICA,. Convolutions IC, * are pro-objects of the category CS.

4.2 Main theorem

By Corollary 3.5.1, the category Dn (G) is a symmetric monoidal category. This allows us to define for any object A E Db(G) objects AkA. We are ready to state the main theorem of this Chapter:

Theorem 1.

a) AkSprp ~ (ICk @ ICDAk+), for 0 < k < n.

b) AnSprP ~CIAn

c) ICA * 6 is invertible under convolution.

d) An+ 1Sprp = 0.

Corollary 4.2.1. There is a monoidal functor z : Rep(G) bG, sending V, to Sprp * 6, and a monoidal functor

Sc o -z : Rep(G) -+ bf,

33 sending representationsof GLn to central objects.

The proof will proceed by induction in n. When n = 1, Sprp = 51 - a sheaf supported on 1 E C* with a 1-demensional stalk, and we need only the fact that A26, = 0, which is obvious. Assume that the theorem is known for groups GLk, k = 1,..., n - 1.

Proof of a). We will now compute AkSprp from the known answer for GLk. Let Pk -> B be a parabolic subgroup of operators fixing a given k-dimensional subspace Vk C V, and let Lk ~_GLk x GLnk be its Levi subgroup, regarded as a two-block subgroup of GL,. Let G' = GLk = GL(Vk), let P' be a parabolic subgroup fixing a vector in P(Vk). We have a corresponding parabolic Springer sheaf Sprp, on G'. Let Sprk = SprP' 6 1, a perverse sheaf on Lk. Recall that lrpk : Pk - Lk denotes the standard projection. For a line 1 = xP E G/P ~_P(V), let U = xUpx-. Consider the following variety:

Sp = {(l, g) E P(Vk) x GLn : g E U1}.

We have the natural projection rR, : Sp -+ Pk. Define

_F1 = 7r _CR, [2 dim Up],

F2 = 7w*pSprk [2 dimUp]. Note that, since centralizers of all elements in Pk are also connected, it makes sense to talk about Schur functors with respect to the convolution in the category D b(Pk), see Corollary 3.5.1. Recall notations from Section 3.3.

Lemma 4.2.1. iAkF1 ~ AkSprp.

Proof. We will regard elements of G/Pk as elements of the Grassmannian Gr(k, n) of k-dimensional subspaces in V. We have the following convolution varieties for r, i k and Spr4:

S.{(l,. ,l k,g, ... k,7 V') E P(Vn)k x Gk x G/Pk : gi E U1 , li C V'},

CA= li,...,i lk,gi,7 ...,g ) n(V)k x G k : g, E Uli}, and a commutative diagram Ok Ck(4.1)

G

Here f is the map forgetting the k-subspace V', and mk is the composition of the projection to Gk and multiplication (gi, ... , g,) - 91 ... Ak.

34 We have

ijNF ~~ m _k!Cok [2k dim Up] ~i mk f!Ck [2k dim Up],

Spry ~ mkCck [2k dim Up]

Since f is proper and an isomorphism over an open set where {l1,... , l} are in general position, f!Cck is a semi-simple complex, containing CC as a direct summand. We get that

AkSprp C AF ~' 0 k~I~~

Now, since hypercohomology with compact support is a monoidal functor, we can compute it for AkSprp and get that the hypercohomologies of the sides of the above inclusion are the same, so it is an isomorphism.

Lemma 4.2.2. a) AkF1 AkF 2 .

1 b) Ak+ F1 = 0.

Proof. Assume that we know that the lemma is proven for all ko < k. We will now prove it for k.

It is easy to see that Upk-averaging of T1 and F2 is the same, namely

WP!F1 XP!F2,W

and, since rPk,! is monoidal,

TPk! A T1 Apk!7 Ak r2.

Since F2 is Upk-equivariant, we have

AkF2 ~2-7pApk',! A F2 [2 dim UQ]

Consider a variety

C (Il =(,..., Ik,91,...,g9k) EP(V)k x GA : g, EU}

We have a morphism

m : C -+ P, (l,..., lk,1g, ... gk) -+ g1 ... gk, and mk!CCk [2k dim Up] = 1l Consider an open subset ,k of Ck defined as a subset of (li, ... ,qlg ... ,7gk) with (li, ... ,lIk) in general position, and let Zk be its complement in Ck. We denote by Mk, abusing notation, the restriction of mk above to Uk, Zk. We have a distinguished triangle

(nk!_Uk, F*, mk!CZk). (4.2)

35 It is easy to see that the complexes mk! k, mk!CZk inherit the Sk-action, compatible with the Sk-action on .*Fk coming from the braiding. We now claim that mk!k is Upk-equivariant. For this purpose, we construct a map a, making the following diagram commutative:

Uk X Up" Uk

tMk Pk

Here m' is the composition

Uk x Up -kMkxl > Pk, xUPk - + Pk,

Choice of a Levi subgroup Lk of Pk gives a splitting V = Vk E W. Given such a splitting, Up, is identified with the subgroup Id + Hom(W, Vk). Any splitting

V = 1' E) -.-) 1', dim 1/ = 1, defines a splitting

UQ = (Id + Hom(W, li)) x ... x (Id + Hom(W, l')), with (Id + Hom(W, l1)) c Up.

Pick a point (l,..., l, gi 1..., gk,u) E Uk x Upk. Let

Rj = gi+19i+2 - Y k,

1 1' - R-' j. Note that, since (1i) are in general position, so are (1'), since the transition matrix in PGLk between these collections is triangular. Let

be a splitting of u corresponding to the splitting (l) of Vk, as above. Let

ti = Rju'R-1.

Note that ui E U1 = UR,,P. We set

,. , i, . , , u) =(, . ,U, I U, .. , gkUk).

It easy to see that g, . .. gkU = g1U1 ... gkk, so the desired map a is now constructed. It remains to observe that a defines a free action of Up, on Uk, so that we have

mk _Uk XUpk [2 dim Upk] ~ m ,k!k

36 so the latter sheaf is indeed Ups-equivariant. Recall the distinguished triangle (4.2). Applying the projection V to the isotypical component of the sign representation to it, we get a triangle

(0mk xk, AkE 1 , V)mkiCZk).

First complex in this triangle is Up,-equivariant, so, to prove part a) of the lemma, it is enough to show that

1 Ak+1ly _k+ F 2 o.

By our inductive assumption, we want to prove that AkSprp ~ Ind' (Ak Sprk), which follows from the above two lemmas.

For a parabolic subgroup Q D B with a Levi L, we say that a weight A of Lie T is Q-regular, if its stabilizer in the full Weyl group W is equal to W(L). Lemma below and its use in the proof are due to Roman Bezrukavnikov.

Lemma 4.2.3. Let the weight A of T be Q-regular. Then for any T E Db(G),

OCQ(F) * ~:X (i o Res (F)) *

Proof. (see Section 3.3 for notations) By definition, we have iL o Res (F) = 6 Q*SJI cQ(Y). Let j YQ\DQ - YQ be the complementary open embedding. We have a distinguished triangle

(jij~cQ(F),t)CQ(T), ZL o Res G(_F)).

6 We will now prove that jj* cQ(F) * A 0. We have

cQ(jijTOCQ(T) * 6,) = j* AtA where j' is an embedding of an open subset V = {(xU, yU)T E Y, x-1 y V Q}, and g Dn(V). Let f be the preimage of V in Y x Y. V has a filtration by locally closed subsets V. = {(xU,yU),x- 1 y c Ug.Q}, where w runs through W- for a subset J of simple reflections in W, corresponding to Q, and gw are some lifts of w to ZG(T). Let j' V: -+ V be the corresponding locally closed embeddings. Consider j'g as a T-equivariant complex on Y x Y. By Lemma 3.2.1, j/9 E (j1,,j1(9).wcw, so it is enough to prove that X '*g * A = 0. Consider a point (xU, yU) E Vt,. Choose

37 X, y in the corresponding congruence classes, so that x-y = gl, 1 E L. Since A is Q-regular, there is a coweight p'V : Gm - T with KA, pv - w(uv)) 7 0, and such that

It(Gm) is in the center of L. Then, for t E Gm, we have

Ad(pv(t))(x-1y) = pv(t)gwlpv(t)-' = pv vw~p(t))-1(x-1y), so that the G-equivariant sheaf, equivariant with respect to the right diagonal T- action, is also equivariant with respect to the action of Gm defined by the coweight (pV - w(,uv)) on one of the components. By Lemma 3.6.1, we get the result. El

Note that since Sprp = IX E IC,\, we have

AkSprp - k ICA2 E Ak-1ICA 2

Proof of c). By the result of [BFO12I, the functor Vi * [c(-) commutes with the Verdier duality. We get that

Yi D(VW1 * c(6)) ~- VW, * c(DW), so that [c(ID) ~- A * A'i by Lemma 3.4.1, which is invertible. But 6 is the IC- extension of the shifted local system S on a set of regular semisimple elements of G, dual of which is isomorphic to sgn 0S, where sgn is the local system corresponding to the sign representation of S,. It is easy to see that IC-extension of sgn 08 is IC\n *6, so the latter is invertible. l

Proof of d). This is equivalent to A"IC 2 = 0. By induction and by Lemma 4.2, we get that (c(A'ICA,) * a i 0 for all A # 0. So, by Corollary 3.6.1, A"IC 2 is a unipotent character sheaf. On the other hand,

A"IC9 ~ AnIC 2 * CT A"-ICA * C * 6 IC 2 * ICn * S ~C * D. and so 1 A"JCA2 * (D6)- C IC 2 * 6. But it is easy to see that the last complex is perverse and indecomposable, so

A"CA 2 = 0, and we are done. 1:1

Proof of b). This now follows from a) and d). r-

4.3 Lie algebra version of the main theorem

The proof above can be straightforwardly adopted to the case of the sheaves on the Lie algebra gn of GL,, as in [Lus87I, [Mir04I. Identifying the nilpotent cone K with the unipotent cone Af,, we get perverse sheaves ICA on g In, as in Section 4.1.

38 Category Db(g(r) is equipped with convolution with respect to the additive group structure on grn. Since this group structure is abelian, Db(g(n) is naturally a sym- metric braided category. Let sp-c E Db(g[n) be the perverse sheaf ICx1 E IC,\. We have the following

Theorem 2. a) AkSprp ~ (ICA e ICAk 1), for 0 < k < n.

b) A IC

c) An+lsJp = 0.

Proof. The proof of part a) is the straightforward adaptation of the proof for the Lie group. So only the part c) needs a separate proof. In the Lie algebra case it is much simpler. Recall the Fourier transform functor

FT : D',(9(n) -+ D%(O[n) from [Lus87], [Mir041. Here D b(g(n) stands for the derived category equivariant with respect to the contracting action of G, on g(. FT intertwines the convolution monoidal structure with the regular tensor product. Let gI,,P - {(xIl) E g[7 x P(V) :l c l} be the parabolic Grothendieck-Springer variety. We have the natural projection

and FT(sprp) = ,_(C [n2]

But all the fibers of 7r are unions of projective spaces having the total dimension of cohomology bounded by n. So, with respect to the monoidal structure given by tensor product, we have A n+I C ~ [n2 and so An+lSptp = 0 with respect to the additive convolution. -

4.4 Another description of the isomorphism in the main theorem

We will need the following description of the isomorphism

AkSprp ~ ICk (EICAk+1'

39 Let L be a Levi subgroup of P. We have the natural morphism of functors

Sprp * ( Xp1cp(-) -+ Xpop** Acp(-) = IndG ResG(-), and so a morphism #4 - AkSprp -+ (IndG Res2)k ('CA) (4.3) Lemma 4.4.1. $k gives an isomorphism to the corresponding summand of the right hand side of (4.3).

Proof. For k = 1 the statement is obvious. Assume that it is known for k = ko. We have a morphism Sprp * AkoSprp -+ Ind ResG (AkoSprp). We want to show that, restricted to the corresponding direct summands, it gives an isomorphism. But the morphism above gives an isomorphism of the stalks at the unit e E G, and the summands in question are semi-simple perverse sheaves with all irreducible summands having non-zero stalks at e, so we are done. E

40 Chapter 5

Extension to the functor from the perfect derived category

Since we have G = G, in this section we will omit the Langlands duality from all notations. Define Jucys-Murphy sheaves Li to be

L, = Af' , l L-_1 *Ar_, where si = (i i + 1), cf. Chapter 2.1. For a weight A (A, ... , A,) of GL" write

LA = L *A2 *---*L*A". L*2 n

Note that we have LA * L, ~ LA+,, since the fact that Jucys-Murphy elements commute follows just from the braid relations. In this section we will prove

Theorem 3. There is a monoidal functor : DGf(St) -+ Db satisfying the following:

a) z(Qg,) - '. 0.

b) z(Og(A, [t)) = L A*'wo * L, for any two weights A, yi of GLn.

c) zu(V 9 Ost) = ( c o =z(V)) * To for any representationV of G.

Remark. This theorem can be thought of as a categorification of the projection flperf, see Section 2.3.

Proof. We employ the general construction of [Bez16. Let t be the Lie algebra of T. Let Cb be the preimage in g x Y of j under the natural projection g x Y -+ g x B. Since the right T-action on Y is free, we have CohGxT(C,) GCoh(b). Let V = Spec I'(Oy) be the affine closure of Y. Now pick a representation V of G, such that

a) V is a multiplicity-free sum of irreducible representations;

41 b) Y is a locally closed subvariety of V, namely an orbit of a highest-weight vector;

c) Y is a closure of Y inside V.

Choose an action of t on V so that t C t acts on an irreducible summand with the highest weight A by a constant (A, t). Let Cb be a closed subscheme of g x t x Y given by

C = {(x, t, v) E g x t x Yc g x t x V: x(v) = t(v)}.

Cb becomes an open subset in Cb, given by (x, t, v) with v having a non-zero projection to all irreducible summands of V. Let Cst be the preimage of the diagonal under the map C6 x C -+ g x g. We have an open subset Cst C Cst with a free T2 -action, and Cst/T2 St By an action of a monoidal category A on a triangulated category B we mean a monoidal functor a : A - End(B), with all a(x) respecting the triangulated structure. For a variety X with an action of an algebraic group H, let Coh .(X) be the full subcategory of CohH(X), consisting of objects of the form V 9 Ox, V E Rep(H). We have the following Proposition ([Bez16], Corollary 18 and Chapter 4.4.1). Let C be a C-linear additive monoidal category. Suppose we are given

a) A tensor functor F : Rep(G x T) -+ End(C), respecting the triangulatedstruc- ture.

b) A tensor endomorphism E of FRep(G), satisfying

Evlgv2 = Ev1 0 IdF(v2 ) + IdF(v1 ) & Ev-

c) An action of O(t) on F by endomorpisms, so that for f E O(t) we have fv 0 v= fv 1 0 Idv2 =Id 1 & fv2 .

d) A "lowest weight arrow" wx : F(V,,0 ) -+ F(A) making the following diagrams commutative: F(VwO 0 Vw 09 ) > F (Vwo(+t))

F (A) (c F(p) ~>F(A + p)

F(Vx) w-t F(A)

F(Vw,xs) w" > F( A)

42 where the right vertical map is the action of the element A C t C 0(t) coming from c). Then the tensor functor F extends uniquely to an action of Coh", (Z7) on C. If we have two copies of the above data, we get an action of Coh7G2xT2 (Q x Qb). In this case, assume, moreover, that e) the action of Rep(G x G) factors through the diagonal G -÷ G x G. f) this action of Rep(G) is by product with central objects in a braided category C, with endomorphism E from b) respecting the central structure.

Then we have an action of Coh ,*T 2(Cst), extending the above data. We now indicate the structures listed in the above proposition, constructing an action of CohWT 2 (s) on Df. First, the action of Rep(G x G) is given by restriction to the diagonal and con- volution with central sheaves from Corollary 4.2.1. Action of Rep(T x T) is given by left and right convolutions with Jucys-Murphy sheaves LA. Action of (f 0 g) E 0(t x t) is given by the action of f and g coming from left and right monodromy, respectively. We construct the rest of the data and prove the needed compatibilities in the following sections.

5.1 Lowest weight arrow and monodromy endomor- phism

For k < n consider the standard representation V of GLk= GL(Vk). Let trivGLA_ stand for the trivial representation of GL,-k. Consider

(k - z(A k k) MZz(trivGLf), a (pro-)character sheaf on the Levi subgroup Lk = GLk x GLnk of G = GLn. Here the ligature q; stands for "Steinberg". Note that we abuse notation here, denoting by =z functors G = GLI with different 1. It should always be clear what we mean. We have -z(An) = Ind(q;) C; is the IC-extension of the local system on the set of regular semisimple el- ements G x GL, given by a module over the extended affine Weyl group (Sk X Sn-k) X Zn, with a pro-unipotent action of Z'. Or, passing to the logarithms of the monodromy, by a module over

Ak = C[Sk X Sn-k] X S(t) = C[Sk X Sn-k] X C[ei,..., en].

Note that the action of (ei + - + ek) E Ak is by a module endomorphism. Let Pk be a parabolic subgroup corresponding to Lk. We have

fc o =z(AkVn) =Ci o b), oXyki,(4k).

43 Let wk be the lowest weight of AkV . We have

,q)eLo =Li* ---*L, BPk 0oiLk (4k) - lwk=L*.*k,

see the proof of Theorem 1. We also have the natural morphism

rk : [CPk XPk iL,(Gk) --+ ik L(;k),

see Section 3.3. Combining, we get an endomorphism

Indik(ei + --- + ek) =: E End(vz(AkVn)), which makes the following diagram commutative:

wAkVC " .

Ekj {Wjk

Here w, - CM(rk), Ek = (Ek), and wk comes from a monodromy action on Lk . We define wwk to be the lowest weight arrows for the images of fundamental representations AkVn. Note that, since the lowest weight vector of any representation can be given by a product of the lowest weight vectors of fundamental representations in their tensor product, we may define, for a lowest weight A = , w, = JJi W*. Now the condition b) of Proposition 5 recovers Ev from E1 in a unique way. The only thing left to check is

Lemma 5.1.1. Elk obtain from E1 by imposing condition b) is equal to Ek defined above.

Proof. Recall that we have a natural morphism of functors

Sprp * (-) - IndG ResG(-),

which gives a morphism

Spr*(-) - (Indp Res )k() By the result of Section 4.4, it gives an isomorphism on the direct summands corre- sponding to the exterior power. If a character sheaf F is given by the IC-extension of a local system corresponding to a C[Sn] x Z-module M, then IndG ResG F is given

by the IC-extension corresponding to a C[Sn] V Z -module C[Sn] Oc[Sn 1 M. Let V be the n-dimensional permutation representation of Sn. Choosing an iso- morphism of Sn-modules

C[S] lC[Sn-k1 ®c[s_1J trivs. ~ V*k

44 and comparing the actions under this identification, we get that the action of E,, on. the corresponding direct summand is the same as given by Ek. l

5.2 Filtration by Jucys-Murphy sheaves

We have that Pk -> B is a parabolic subgroup corresponding to the subset J {si,.. . , s_ sk+1, ... , s1} c I. By Lemmas 3.1.1, 3.2.2 and the discussion in Section 2.2, we get

Lemma 5.2.1. ( c o =z)(A V EZ) (L\x)>ewwk.

5.3 Passage from Cst to St

We constructed the action of Coh 2 CStsT on bf. Consider the action on the object TWO. This gives a functor 7: CohGx 2Cst -- bf. Since Rep(G x T 2) act by convolution with objects having a filtration by products 2 of tS, discussion of tw. in Section 3.4 gives that 7 in fact sends Coh jT Cst to T. Thus, we get a functor

2 Ho(CohGxT Cst) --+ Ho(!T) ~ bf.

Finally, we employ the following proposition from [AB091. In any tensor category over a field of characteristic zero one can construct a Koszul complex KO,d associated to any morphism 0: V -+ L and d E Z ", namely the complex

0 - Ad (V) - Ad-I(V) 0 L -+ -.. Ad-i (V) 0 Sym'(L) -+ --- - Sym d(L) -+ 0.

Let KA be the Koszul complex Kwx,dimVA in (Coh GxT2uSt, 0o

Proposition 1 ([AB09J, Lemma 20). Assume there is an action of

2 a: Ho(Coh rxT Cst) ' C such that a(KA)x = 0 for all A and some object x E Ob(C). Then the functor

P : Ho(Cohx y2Zst) -+ C, y +a (y)x factors through the functor F:Derf(St) -+ C. Functor T satisfies the condition of the above proposition, because, on an object of the form V® 0 , V E Rep(G x T2), it is given by V & to . We have

$(KA) = k \ T o = 0,

45 where kA is the (obviously acyclic) Koszul complex in Rep(G x T2 ). Now, as in [Bez161, the fact that the action of E is pro-nilpotent implies that we get a functor 4Df zu : D4(tper: D,($t)G -+b.

5.4 Compatibility with the monoidal structure

It is shown in [Bez16] that the convolution F * g in DGf (St) can be computed by taking total complex of a bicomplex {JF * gi}, where {F2 }, {g } are complexes representing F, g in Ho(CohfrSt). Similarly, the convolution F * g in Df can be computed by taking total complex of a bicomplex {F * gi}, where {P}, {gi} are complexes representing F, g in Ho(T). Thus, we only need to check

7'(At * Os) 'wo * two, which is again a result of op.cit.

46 Appendix A

Remark on the Kirillov-Pak character formula

In this Appendix we discuss a formula from [KP90]. We include it for the sake of completeness, because it was a part of our initial motivation for choosing the parabolic Springer sheaf as the image of the standard representation of GL" in Theorem 1. We guessed it independently using CHEVIE package [GHL+96] for GAP3 computer algebra system [S+971. Let V be the standard n-dimensional permutation representation of Se, and con- sider R = C[x1, .. ., x,] as a graded Sn-module, with Sn-action permuting the variables. Note that, on the one hand, local system defining the character sheaf Sprp * 6 is given by R 0 V considered as a C[S] < R-module. On the other hand, r c(Sprp * 6) caregorifies the sum of Jucys-Murphy elements, and this is known to act in the irreducible module EA of H[ corresponding to a Young diagram A by the constant

qC(E")) YII q C(E~i) - 1 (qc(Ei) where the sum is taken over the boxes of diagram A, c(-) stands for the content of a box, and q = v 2 , see [I005J. We can write the same expressions for AkSprp *6 involving elementary symmetric polynomials ek in Jucys-Murphy sheaves, see 2.2. Let A be the ring of symmetric functions, and let Xq stand for the Frobenius character from representations of S, to A[[q]]. Let SA be the Schur symmetric function corresponding to a partition A. Define power series QA E Z[[q] by

Xq(R) = Z QSx,

The above discussion suggests the following:

47 Proposition ([KP90).

Xq(A kV o R) = E ek (qc(El1 (A)), I ...,q c(]. (A)) ) QAA A

Proof. Indeed, it is well-known that

1 QA = sa(1, q, q2,... ) = qb(A) HEA( h(i))' where b(A) = EZ(i - 1)Aj, and h(.) stands for the hook length, so the proposition is precisely the statement of [KP90] in the coefficient of sk l

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