Categorical Actions from Lusztig Induction and Re- Striction on ﬁnite General Linear Groups

Categorical actions from Lusztig induction and restriction on ﬁnite general linear groups

O. Dudas, M. Varagnolo, E. Vasserot∗

Abstract. In this note we explain how Lusztig’s induction and restriction functors yield categorical actions of Kac-Moody algebras on the derived category of unipotent representations. We focus on the example of ﬁnite linear groups and induction/restriction associated with split Levi subgroups, providing a derived analogue of Harish-Chandra induction/restriction as studied by Chuang-Rouquier in [5].

2010 Mathematics Subject Classiﬁcation. Primary 20C33. Keywords. Finite reductive groups; Deligne-Lusztig theory; Higher representation theory.

Introduction

Let Gn = GLn(q) be the general linear group over a finite field with q elements. In order to construct and study the representations of Gn with coefficients in a field k of positive characteristic ℓ ∤ q, it is common to consider the chain of subgroups {1} = G0 ⊂ G1 ⊂ ··· ⊂ Gn ⊂ · · · together with the Harish-Chandra induction and restriction functors − also called parabolic induction and restriction −

F - kGn-mod m kGn+1-mod. E

But there is more to it: Chuang-Rouquier showed in [5] that for any m ≥ 1, the functor Fm is endowed with a natural action of an affine Hecke algebra of type Am−1 controlling most of the behaviour of these functors. From this action they defined a family of i-induction and i-restriction functors {Fi} and {Ei} inducing, at the level of Grothendieck groups, an action of a Kac-Moody algebra g. This led to the notion of categorical g-action on the category of (unipotent) representations for various Gn (see [18]). Note that for this construction to exist we need to assume in addition that ℓ ∤ q − 1. The existence of a categorical action has many consequences, among them the description of Harish-Chandra series and the construction of derived equivalences. It was shown in [5, 18] that an abelian category C endowed with a g-action is built from minimal categorifications, each of them corresponding to a composition factor of the g-module K0(C). In addition, these minimal categorifications are unique, as

∗This research was partially supported by the ANR grant number ANR-13-BS01-0001-01. 2 O. Dudas, M. Varagnolo, E. Vasserot they are given by the module category of a Hecke algebra of ﬁnite type which can be read oﬀ from the highest weight of the corresponding simple g-module. In this note we consider a similar situation, where Harish-Chandra induction and restriction functors are replaced by Lusztig induction and restriction

F . b mod b mod D (kGn- ) n D (kGn+1- ). E

These functors are defined using the mod-ℓ cohomology of Deligne-Lusztig varieties associated to central elements in the braid monoid (see §2.1 for the definition). They are no longer exact, but triangulated functors between derived categories. Nevertheless, we can show how they fit in a representation datum (see Proposition 2.1). Proposition. Assume ℓ ∤ q − 1. Then there exist endomorphisms X ∈ End(F)× and T ∈ End(F2) satisfing the following relations:

(a) 1FT ◦ T 1F ◦ 1FT = T 1F ◦ 1FT ◦ T 1F,

(b) (T +1F2 ) ◦ (T − q1F2 )=0,

(c) T ◦ (1FX) ◦ T = qX1F. This endows Fm with a natural action of an affine Hecke algebra. Unlike the case of Harish-Chandra induction and restriction, the natural transformations of Fm we consider are of a geometric nature, as they are induced by endomorphisms of the corresponding Deligne-Lusztig varieties. These operators were already consid- ered by Broué-Michel [3] and Digne-Michel [7] in order to understand the structure of the endomorphism ring of the cohomology of Deligne-Lusztig varieties. The ad- vantage of their construction is that braid relations are automatically satisfied. Again, we can define a family of i-induction and i-restriction functors {Fi} and {Ei}. However, in order to show that they induce an action of a Kac-Moody algebra on the Grothendieck group of the category of unipotent representations kGn-umod we need to assume some vanishing properties of the cohomology of Deligne-Lusztig varieties which were conjectured by Broué-Michel [3] (see §2.2 and more specifically Corollary 2.5 for details). Proposition. Assume that the ℓ-adic cohomology groups of the Deligne-Lusztig variety X(π) vanish outside the even degrees. Let e be the order of q modulo ℓ. If e> 1, then the tuple (E, F,T,X) induces a categorical action of sle on

b b b D (kG-umod) := D (kGn-umod). Mn≥0 In that case, the action that we obtain at the level of Grothendieck groups is the same as the one coming from Harish-Chandra induction and restriction, although the functors (and hence the categorical actions) are very diﬀerent. They should however be intertwined by a perverse equivalence. 3

Our construction provides an example of categorical actions on triangulated categories. It would be interesting to study the consequences of the existence of such categorifications, but we shall not do it here. Our motivation comes from similarities between the representation theory of finite general linear groups and finite general unitary groups, often referred to as Ennola duality. Let GUn be the finite general unitary group over Fq2 . There is no Harish-Chandra induction to GUn+1 but there is a Lusztig induction

′ F . b mod b mod D (kGUn- ) n D (kGUn+1- ) E′

between derived categories, which ﬁts again in the theory of categorical actions. On the other hand, Brou´e-Malle-Michel showed in [2] that there is an isometry between unipotent characters of Gn and unipotent characters of GUn which intertwines Harish-Chandra induction and restriction with this Lusztig induction and restriction as shown in the following diagram:

F [ ] . mod mod K0(kGn- ) n K0(kGn+1- ) [E]

∼ F′ ∼ [ ] . mod mod K0(kGUn- ) n K0(kGUn+1- ) ′ [E ]

In other words, the g-representations aﬀorded by the Grothendieck groups of unipotent representations of general linear and unitary groups are isomorphic. Since both representations come from a categorical g-action, we hope that this might lead b b the way to a proof that the derived categories D (kGn-mod) and D (kGUn-mod) are equivalent. Note that the g-representations at the level of the Grothendieck groups are not simple, therefore one would need to consider an additional action of a Heisenberg algebra which can also be categoriﬁed using Lusztig induction and restriction. We hope to carry out this project in a future work.

1. Parabolic Deligne-Lusztig varieties

In this section G is a connected reductive group over Fp and F : G −→ G is a Frobenius endomorphism deﬁning an Fq-structure on G. Given an F -stable subgroup H of G we will denote by H = HF the corresponding ﬁnite group.

1.1. Braid groups. Let B be an F -stable Borel subgroup of G and T be an F -stable maximal torus of B. To the pair (B, T) one can associate the Weyl group W = NG(T)/T and the set S ⊂ W of simple reﬂections. An element s ∈ W lies in S if and only if BsB/B has dimension 1 in G/B. Since B and T are F -stable, F induces an automorphism of W which normalizes S. 4 O. Dudas, M. Varagnolo, E. Vasserot

Let B+ be the Artin monoid corresponding to the Coxeter system (W, S). It is generated by a set of elements S which lift the simple reflections of S. The elements in S have infinite order but they satisfy the braid relations. More precisely, if ms,t denotes the order of st in W then B+ is defined by the presentation

B+ = hS | sts ··· = tst ··· for s, t ∈ Si.

ms,t ms,t | {z } | {z } Since the braid relations are also satisﬁed in W , there is a well-deﬁned quotient map B+ −→ W sending s ∈ S to s ∈ S. In addition, this map has a canonical section + W −→ B sending w = s1 ··· sr to w = s1 ··· sr where s1 ··· sr is any reduced expression of w. The image of this map will be denoted by W. By convention we will always use bold letters to denote elements of B+.

1.2. Parabolic Deligne-Lusztig varieties. Given a subset I of simple reﬂec- tions, we denote by PI the standard parabolic subgroup of G containing B, and by UI its unipotent radical. The group PI has a unique Levi complement LI containing T. The parabolic subgroup of W generated by I, denoted by WI , is the Weyl group of LI . Let w ∈ W . Assume that w is I-reduced, which means that w −1 has minimal length in the coset WI w. Equivalently, w maps any simple root of I to a positive root. Assume in addition that w normalizes I. Then w (and w−1) induces a permutation of the simple roots in I. Then the parabolic Deligne-Lusztig variety associated with the pair (I, w) is

−1 X(I,wF ) = g ∈ G g F (g) ∈ PI wPI PI . This definition can be extended to pairs (I, b) where I is a subset of S and b ∈ B+ are such that b is I-reduced and normalizes I (see [7, §7.1, 7.2]). The corresponding Deligne-Lusztig variety will be denoted by X(I, bF ) or often X(I, b) if F acts trivially on W . In the particular case where b = w is the lift of an element w ∈ W then X(I, bF ) is canonically isomorphic to X(I,wF ). When I = ∅ is empty, any element b of the braid monoid yields a Deligne-Lusztig variety, which we will simply denote by X(bF ), or even X(b) if in addition F acts trivially on W . Following [7, §7.3], we shall also fix an F -stable Tits homomorphism t : B+ −→ NG(T). Then the variety X(I, bF ) has an étale covering X(I, bF ) with Galois b Lt( )F b w group I . When = is the lift of an element w ∈ W thene this covering can be defined by

−1 X(I,wF ) = g ∈ G g F (g) ∈ UI t(w)UI UI . e The varieties X(I, bF ) and X(I, bF ) are endowed with an action of G by left multiplication. The quotient map X(I, bF ) ։ X(I, bF ) is equivariant for this e action. e 1.3. Lusztig induction and restriction. Let ℓ be a prime number diﬀerent from p, and (K, O, k) be an ℓ-modular system such that K is a ﬁnite extension of 5

the field of ℓ-adic integers Qℓ. Given any ring Λ one of (K, O, k) and any finite group H, we denote by ΛH-mod the category of finitely generated left ΛH-modules, and by Db(ΛH-mod) the corresponding bounded derived category. Given any quasi-projective algebraic variety X acted on by H, there are well- b defined objects RΓ(X, Λ) and RΓc(X, Λ) in D (ΛH-mod) representing respectively the étale cohomology and the étale cohomology with compact support of X with coefficients in Λ. In addition, if the stabilizers under the action of H on X are ′ ℓ -groups, then RΓ(X, Λ) and RΓc(X, Λ) are isomorphic to bounded complexes of finitely generated projective ΛH-modules [4, §A.3.15]. These complexes are called perfect. Given (I, b) a pair as in §1.2, the cohomology of the Deligne-Lusztig variety t(b)F X(I, bF ) yields a complex of (G, LI )-bimodules which is perfect as a complex b Lt( )F ofe G-modules and as a complex of right I -modules. Indeed, stabilizers under t(b)F the action of G lie in conjugates of UI , hence are p-groups, whereas LI acts freely on X(I, bF ). Using the cohomology complex RΓc(X(I, bF ), Λ) we can form the following pair of adjoint functors e e R b t(b)F mod b mod I,b = RΓc(X(I, bF ), Λ) ⊗ Lt(b)F − : D (ΛLI - ) −→ D (ΛG- ), Λ I ∗R e b mod b t(b)F mod I,b = RHomΛG(RΓc(X(I, bF ), Λ), −): D (ΛG- ) −→ D (ΛLI - ) called Lusztig induction ande restriction functors. When b = 1 is trivial, they correspond to Harish-Chandra induction and restriction. ∗ When Λ is a field, these functors induce linear maps [RI,b] and [ RI,b] between t(b)F mod mod the Grothendieck groups of ΛLI - and ΛG- , which were originally con- sidered by Lusztig. Note that by [6, Chap. 6] and [7, Prop. 7.7], these maps t(b)F depend only on the finite Levi subgroup LI as soon as the Mackey formula holds, which we know for the groups we will encounter by [1]. In particular, [RI,b] ∗ and [ RI,b] depend only on the image of b in W .

1.4. Action of braid groups. In [3], Broué-Michel defined operators on X(bF ) corresponding to actions of F -cyclic shifts. These were generalized to the case of parabolic Deligne-Lusztig varieties by Digne-Michel in [7]. We will only use them in the following situation: we fix (I, b) as in §1.2 and we assume that we can write b = uv = vF (u) where u commutes with every element of I. In particular, b normalizes I if and only if v does. To this decomposition is associated in [7, §7.4] an endomorphism Du of X(I, bF ) which commutes with the actions of G and t(b)F L u b b I . The particular case of e = gives D = F , the Frobenius endomorphism. The crucial property of these endomorphisms is that if b = u′v′ = v′F (u′) is another decomposition, the endomorphism DuDu′ depends only on the product uu′ in B+. It is denoted by Duu′ even though uu′ is not a prefix of b.

1.5. The central element π. Let w0 be the longest element in W and w0 be + π 2 + its lift to B . Then = w0 is a central element in B . The ℓ-adic cohomology of 6 O. Dudas, M. Varagnolo, E. Vasserot

Deligne-Lusztig varieties attached to π and its roots in B+ is conjectured to have particular vanishing properties. In the case of π, an explicit conjectural description of the cohomology was given in [3, Conj. 2.15] (which was later corrected in [8, Conj. 3.3.24]). Recall that the classification of irreducible unipotent characters is independent of q. The degree of a unipotent character is the evaluation at q of a polynomial with rational coefficients. The degree (resp. the valuation) of the polynomial associated with a unipotent character ρ is denoted by Aρ (resp. aρ). Conjecture 1.1 (Broué-Michel). The ℓ-adic cohomology of X(π) is given by π RΓ(X( ), Qℓ) ≃ ρ ⊗ HomG(ρ, QℓG/B)[−2Aρ] Mρ b mod in D (Qℓ- ), where ρ runs over the set of irreducible unipotent characters. Fur- thermore, the eigenvalue of F on the ρ-isotypic component is qaρ+Aρ .

The constituents of the permutation module QℓG/B are called the principal series representations of G. The conjecture implies that only principal series characters occur in the ℓ-adic cohomology of X(π).

2. π-induction and π-restriction for general linear groups

Throughout this section we will assume that G is a general linear group over Fp, q and F is the standard Frobenius (ai,j ) 7−→ (ai,j ). For the construction of the representation datum to make sense over the ℓ-adic integers, we will assume from now on that ℓ ∤ q − 1. Under this assumption the F ′ group of Fq-points T of any split torus T of G is an ℓ -group.

2.1. The representation datum. Given n ≥ 0, we write Gn = GLn(Fp) and F Gn = Gn = GLn(q), with the convention that G0 = G0 = {1}. We will take Tn (resp. Bn) to be the diagonal matrices (resp. upper-triangular matrices) in Gn. The corresponding Weyl Wn group is isomorphic to the symmetric group. For i =1,...,n − 1, we denote by si the permutation matrix corresponding to the + transposition (i,i+1). The braid monoid associated to Wn will be denoted by Bn . Note that permutation matrices yield a natural F -equivariant Tits homomorphism + t : Bn −→ NGn (Tn) which factors through Wn. Following [5, §7.3], we view Gn−1 as a subgroup of Gn as follows: we first consider the maximal parabolic subgroup Pn of Gn corresponding to the simple reflections {s1,...,sn−2}. It has a unipotent radical Un and a standard Levi complement which is naturally isomorphic to Gn−1 ×G1. If we set Vn = G1 ⋉Un then we have a decomposition Pn = Gn−1 ⋉ Vn. π + For n ≥ 0, we consider the element n of the Artin braid group Bn of Wn defined by induction by π0 = π1 = 1 and

πn = (s1 ··· sn−1sn−1 ··· s1)πn−1

= (sn−1 ··· s1s1 ··· sn−1)πn−1. 7

The element πn is the square of the lift of the longest element in Wn and thus equals the central element deﬁned in §1.5. Note that the image of πn in Wn is trivial. Since πn is central, the element s1 ··· sn−1sn−1 ··· s1 = πn/πn−1 commutes with si for i =1,...,n − 2. With In = {s1,..., sn−2}, the pair (In, πn/πn−1) satisﬁes the assumptions in §1.2 and one can consider the corresponding parabolic Deligne- Lusztig varieties X(In, πn/πn−1) and X(In, πn/πn−1). π π The variety Xe(In, n/ n−1) is endowed with a left action of Gn and a right action of G × G (note that G ≃ F×). Since we will be interested in unipotent n−1 e 1 1 q representations only we will rather work with the variety

Yn = X(In, πn/πn−1)/G1.

It is an ´etale covering of X(In, πn/eπn−1) with Galois group Gn−1. The deﬁnition of Yn can be made more explicit by considering subvarieties of Gn/Vn instead of Gn/Un. More precisely 2 −1 −1 2 Yn ≃ (g,h) ∈ Gn | g h, F (g) h ∈ Vn(1,...,n)Vn Vn. Given r

Yn,r = Yn ×Gn−1 Yn−1 ×Gn−2 ×···×Gr+1 Yr+1. π π × n−r It is canonically isomorphic to the quotient of X(Ir+1, n/ r) by (Fq ) . The variety is endowed with a left action of Gn and a right action of Gr, and therefore b opp mod e RΓc(Yn,r, Λ) ∈ D (ΛGn × Gr - ). In addition, since we assume that q − 1 ∈ × Λ , it can be represented by a complex of bimodules which are projectiveasΛGn- modules and as ΛGr-modules. Using this complex we can form the following pair of adjoint functors F b mod b mod n,r = RΓc(Yn,r, Λ) ⊗ΛGr − : D (ΛGr- ) −→ D (ΛGn- ), E b mod b mod n,r = RHomΛGn (RΓc(Yn,r, Λ), −): D (ΛGn- ) −→ D (ΛGr- ).

Note that Fn,r ≃ Fn,n−1 ··· Fr+1,r and En,r ≃ En,n−1 ··· Er+1,r. Since these functors are deﬁned by bounded complexes of (Gn, Gr)-bimodules, any endomorphism of the variety Yn,r which commutes with the left action of Gn and the right action of Gr yields an endomorphism of the bimodule RΓc(Yn,r, Λ) and therefore a natural transformation of Fn,r (and of En,r). Consequently, we can use the braid group operators deﬁned in §1.4 to construct endomorphisms of Fn,r and En,r. We will consider two cases.

• The element πr+1/πr obviously commutes with itself, and the correspond-

ing operator Dπr+1/πr on Yr+1,r = Yr+1 is induced by the Frobenius endo-

morphism. We still denote by Dπr+1/πr the endomorphism induced on the F −r functor r+1,r and we set Xr = q Dπr+1/πr .

• The element sr+1 centralizes Ir+1 = {s1,..., sr−1}, and in particular it com- π π + mutes with r. In addition, since r+2 is central in Br+2, it commutes with π π r+2/ r. Consequently, there is a corresponding operator Dsr+1 on Yr+2,r which induces an isomorphism on the cohomology of Yr+2,r. We denote by Tr the corresponding endomorphism of Fr+2,r. 8 O. Dudas, M. Varagnolo, E. Vasserot

Let us deﬁne the category

ΛG-mod = ΛGr-mod. Mr≥0

Then the functors F = Fr+1,r and E = Er+1,r form a pair of adjoint endo- functors of Db(ΛG-modL). They are a particularL case of Lusztig induction and restriction functors. Note that when Λ is a ﬁeld, the maps [F] and [E] induced on Grothendieck groups coincide with the Harish-Chandra induction and restriction maps since the image of πn in Wn is trivial. However the functors F and E diﬀer from the Harish-Chandra functors, even when Λ = K ⊃ Qℓ. Setting X = Xr 2 and T = Tr we obtain endomorphisms X ∈ End(F) and T ∈ End(F ).L By adapting theL argument in [16] and [3] one can show the following. Proposition 2.1. The tuple (E, F,T,X) is a representation datum on ΛG-mod. In other words, the endomorphism X ∈ End(F)× and T ∈ End(F2) satisfy the following relations:

(a) 1FT ◦ T 1F ◦ 1FT = T 1F ◦ 1FT ◦ T 1F,

(b) (T +1F2 ) ◦ (T − q1F2 )=0,

Proof. We start by (a). The endomorphism 1FT (resp. T 1F) of F3 is induced by the operator Dsr+1 (resp. Dsr+2 ) on Yr+3,r. Therefore the relation (a) follows directly from the braid relation Dsr+1 Dsr+2 Dsr+1 = Dsr+2 Dsr+1 Dsr+2 (they are both equal to Dsr+1sr+2sr+1 ). The same argument, with the equality

sr+1(πr+1/πr)sr+1 = sr+1(s1 ··· srsr ··· s1)sr+1

= sr+1(sr ··· s1s1 ··· sr)sr+1

= πr+2/πr+1 + in Br+2 yields the relation (c). The relation (b) was shown to hold for the non-parabolic variety X(πr) in [3, Thm. 2.7] but it can be generalized to our setting. We fix r ≥ 0 and we consider I = Ir = {s1,..., sr−1} and J = Ir ∪{sr+1}. We will write for short π = πr+2. opp Then by [7, Prop. 7.19], there is a natural isomorphism of Gr+2 × LI -varieties π π π π ∼ π π X(J, / J) ×LJ XLJ (I, J/ I) −→ X(I, / I). π π Through this isomorphism,e the braide group operator De sr+1 on X(I, / I) is Id × Ds by [7, Prop. 7.24]. Since L ≃ G × G and πJ/πI = s s , the r+1 J r 2 e r+1 r+1 π π variety XLJ (I, J/ I) is isomorphic to Gr ×XG2 (ss), where s is the unique simple reflection of the Weyl group of G2 = GL2. If we mod out by the torus of G2 viewed e × 2 e as diag(1,..., 1,x,y) ≃ (Fq ) then we obtain finally π π ∼ X(J, / J) ×G2 XG2 (ss) −→ Yr+2,r.

In addition, the actione of Dsr+1 on Yr+2,r corresponds to the action of Ds on

XG2 (ss) via this isomorphism and we conclude using the following Lemma. 9

× Lemma 2.2. Recall that q(q − 1) ∈ Λ . The braid group operator Ds on Y1 =

XG2 (ss)= X(ss) induces an endomorphism of RΓc(Y1, Λ) satisfying

(Ds + 1)(Ds − q)=0.

Proof. When Λ = K ⊃ Qℓ, the relation holds by [3, Lem. 2.14]. Here we shall work with Λ = O ⊃ Zℓ. We ﬁrst prove that under the assumptions on ℓ, there is no torsion in the cohomology of Y1. Since it is an irreducible variety of dimension 3 2, the torsion part can only appear in Hc (Y1, O). We start by looking at the variety Y1 as an open subvariety of X(ss) = X(ss) ⊔ X(s) (see [8, §2] for the notation s). By [8, Prop. 3.2.3] and the fact that X(s)= X(s) ⊔ X(1) ≃ P1, there is a distinguished triangle

RΓc(Y1, O) −→ RΓc(P1, O)[−2](−1) −→ RΓc(X(s), O)

b in D (OG2-mod) which is equivariant for the action of F . Let C be the sum of 2 the generalized eigenspaces of F on RΓc(Y1, O) for the eigenvalues 1 and q and let D be the q-generalized eigenspace of F . We get distinguished triangles 1 C −→ O[−4] −→ Hc (X(s), O)[−1] and D −→ O[−2] −→ O[−2] b in D (OG2-mod). Assume that D 6= 0. Then the map f : O −→ O occuring in the previous complex is not an isomorphism, therefore its ℓ-reduction f must be zero. Recall that the complex D is perfect since ℓ ∤ q − 1. Consequently f induces an isomorphism in the stable category kG2-stab, which forces the trivial G2-module 3 3 k to be projective. This holds only when ℓ ∤ |G2|. Now H (D) = H (Y1, O) is a torsion module with trivial action of G2. This contradicts [8, Prop. 3.3.14] and proves that D ≃ 0. In particular the cohomology of Y1 is torsion-free.

b To ﬁnish the proof it is enough to show that EndD (OG2-mod)(C) is torsion-free, b since in that case it will embed into the K-algebra EndD (KG2-mod)(KC) in which Ds satiﬁes the relation (Ds + 1)(Ds − q) = 0 by [3, Lem. 2.14]. To do so we give two explicit representatives of C as a bounded complex of G2-bimodules. We distinguish two cases: if ℓ ∤ q + 1, then ℓ ∤ |G2| and

1 C ≃ Hc (X(s), O)[−2] ⊕ O[−4].

1 Here Hc (X(s), O) is a lattice for the Steinberg character, which is a projective b OG2-module under the assumptions on ℓ. We deduce that EndD (OG2-mod)(C) ≃ (O)⊕2 as an O-module. In the case where ℓ | q + 1, we use the explicit structure of projective indecom- posable modules in the principal ℓ-block of G2 to get

C ≃ 0 −→ PL −→ PL −→ Pk −→ 0 where L is the non-trivial simple module in the block, and PL, Pk are lattices lifting the projective covers of the simple modules L and k. One can check that 10 O. Dudas, M. Varagnolo, E. Vasserot

b HomD (kG2-mod)(kC,kC[1]) = 0. Since C is perfect, we can invoke the universal 0 b coeﬃcient theorem to conclude that EndD (OG2-mod)(C) = H (REndOG2 (C)) is torsion-free.

Remark 2.3. Note that when ℓ | q+1, the cohomology complex C ≃ RΓc(X(ss), O) can be obtained as the image of the projective module Pk by a perverse equivalence with ﬁltration {k}⊂{k,L} and (decreasing) perversity function k 7−→ 4 and L 7−→ 2. This also holds when ℓ ∤ q + 1 but it is less interesting since in that case the underlying categories are hereditary.

The relations satisfied by X and T correspond to relations in affine Hecke q algebras. For m ≥ 1, the affine Hecke algebra HΛ,m is the Λ-algebra generated by ±1 ±1 T1,...,Tm−1, X1 ,...,Xm subject to the relations

• Type Am−1 Hecke relations for T1,...,Tm−1:

(Ti + 1)(Ti − q)=0,

TiTi+1Ti = Ti+1TiTi+1 and TiTj = TjTi if |i − j| > 1,

±1 ±1 • Laurent polynomial ring relations for X1 ,...,Xm :

−1 −1 XiXj = Xj Xi and XiXi = Xi Xi =1,

• Mixed relations:

TiXiTi = qXi+1 and XiTj = TjXi if i − j 6=0, 1.

Then given (E, F) a pair of biadjoint functors together with X ∈ End(F) and T ∈ End(F2), the tuple (E, F,X,T ) is a representation datum if and only if for each m ∈ N, the map

q Fm φFm : HΛ,m −→ End( )

Xk 7−→ 1Fm−k X1Fk−1 for all 1 ≤ k ≤ m Tl 7−→ 1Fm−l−1 T 1Fl−1 for all 1 ≤ l ≤ m − 1 is a well-deﬁned Λ-algebra homomorphism.

2.2. Fock space representation. Following [5, §7], we can consider the generalized eigenspace of X on F and E. Since X and T satisfy relations of an affine Z Hecke algebra, the eigenvalues of X all lie in q . For i ∈ Z, we define Fi and Ei to be the qi-generalized eigenspaces of X on F and E (see Remark 2.9). Then by definition E = Ei and F = Fi. For each i ∈ Z, the functors Fi and Ei induce linear mapsL on the GrothendieckL group of Db(ΛG-mod) which we denote by [Fi] and [Ei]. We would like to show that the linear maps [Ei] and [Fi] for i ∈ Z induce an action of a Kac-Moody algebra g on K0(ΛG-mod). This will endow Db(ΛG-mod) with a structure of categorical g-representation. To this end we will 11

work under two restrictions: we will only work with unipotent representations, and we will assume that Conjecture 1.1 holds. Assume that Λ is a ﬁeld, one of K or k. A simple ΛG-module is unipotent if it occurs as a composition factor of Hj (X(w), Λ) for some w ∈ W and j ∈ Z. We denote by ΛG-umod the Serre subcategory of ΛG-mod generated by unipotent simple modules. It is a direct summand of ΛG-mod. By [9, 10], the simple unipotent ΛGn-modules are naturally parametrized by partitions of n such that the decomposition map

d : K0(KG-umod) −→ K0(kG-umod) is unitriangular with respect to the dominance order on partitions. In particular, it is an isomorphism of abelian groups. Given λ a partition of n, we will denote by ∆λ (resp. Lλ) a simple KGn-module (resp. kGn-module) corresponding to λ. For example, ∆(n) and L(n) correspond to the trivial representations over K and k respectively, whereas ∆(1n) is the Steinberg representation. Note that by [14] any irreducible unipotent representation over K (resp. over k) is actually deﬁned over Qℓ (resp. over Fℓ), and is absolutely irreducible over this ﬁeld.

Recall that to a partition λ = (λ1 ≥ λ2 ≥ ··· ≥ λr) one can associate its Young diagram Y (λ) = {(x, y) | 1 ≤ x ≤ r, 1 ≤ y ≤ λx}. The content of a node (x, y) ∈ Y (λ) is y − x. We say that a node of content i can be added to λ if there is a partition µ of n + 1 such that Y (µ)= Y (λ) ∪{(x, y)} with y − x = i. If such a partition µ exists it is unique and we write µ = λ⋆i.

Proposition 2.4. Assume that Conjecture 1.1 holds. Let λ be a partition and i ∈ Z. Then

0 if one cannot add a node of content i to λ [Fi]([∆λ]) = [∆λ⋆i] otherwise where λ⋆i is the partition obtained from λ by adding a node of content i.

Proof. Assuming that Conjecture 1.1 holds, we can compute explicitely the ℓ-adic cohomology with compact support of X(πr) with the action of Gr and F . A unipotent representation ∆µ corresponding to a partition µ of r occurs in degree r(r−1)−aµ−Aµ 2r(r−1)−2Aµ and with eigenvalue q , with aµ := a∆µ and Aµ := A∆µ .

Let λ be a partition of n. By [7, Prop. 7.19] and the deﬁnition of Yn+1, we have π π Yn+1 ×Gn X( n) ≃ X( n+1) from wich we get, by taking the cohomology

F(∆λ) ≃ ∆µ[2n(n − 1) − 2Aλ − 2n(n +1)+2Aµ] |µ/λM|=1

≃ ∆µ[2(Aµ − Aλ − 2n)] |µ/λM|=1 12 O. Dudas, M. Varagnolo, E. Vasserot where the sum runs over all partitions µ of n + 1 which are obtained from λ by adding one node. In addition the eigenvalue of F on ∆µ is the quotient of the eigenvalue on ∆µ in the cohomology of X(πn+1) by the eigenvalue on ∆λ in the 2n+aλ+Aλ−aµ−Aµ cohomology of X(πn). This equals q . Now by deﬁnition, the action of X coincide with the action of q−nF , giving, for i ∈ Z

Fi(∆λ) ≃ ∆λ⋆k[2(Aλ⋆k − Aλ − 2n)]. (1) Mk∈Z n+aλ+Aλ−aλ⋆k−Aλ⋆k=i

Now, for a partition λ = (λ1 ≥···≥ λr) of n, Lusztig’s a and A-functions are given by

aλ = λl 1≤Xj

By adjunction, we deduce from the proposition that the action of [Ei] on unipotent characters corresponds to removing nodes of content i. These actions on partitions are well-known, and they deﬁne a structure of slZ-module on the space generated by partitions, called a Fock space representation of level 1. This was previously known for Harish-Chandra induction and restriction, see for example [15] for the case of Hecke algebras of symmetric groups. Using the decomposition map and the fact that F and E are deﬁned over O, we obtain a similar result for the positive characteristic case.

Corollary 2.5. Assume that Conjecture 1.1 holds. Let e be the order of q modulo ℓ. If e> 1, then

[Fi] d([∆λ]) = d([∆λ∗(i+je)]). Xj∈Z Note that the assumption e> 1 is equivalent to q(q − 1) ∈ Λ×, which is needed for the construction of the functors E and F.

Remark 2.6. In the proof of 2.4 we only used the assumption that a given unipotent character occurs in even degree cohomology groups of X(πr) only, and with a given Frobenius eigenvalue. We did not actually use that this degree should be unique and related to Lusztig’s A-function. Consequently, for Proposition 2.4 and its Corollary to hold, it is enough to assume by [3, §2.16] that the odd-degree cohomology groups of X(πr) vanish. 13

This shows that K0(kG−umod) has a natural structure of sle-module, which is again a Fock space of level 1. As before, this was already known for Harish- b Chandra induction and restriction functors. This generalize the property that Harish-Chandra induction and restriction coincide with F and E at the level of Grothendieck groups. Motivated by Brou´e’s abelian defect group conjecture, we can expect the following stronger statement to hold. Conjecture 2.7. Let (E, F, T, X) be the representation datum associated with Harish-Chandra induction and restriction in [5, §7.3]. There exists a derived self- e e e e equivalence of Db(ΛG−umod) which intertwines (E, F, T, X) and (E, F,T,X). Furthermore, this equivalence should be perverse,e e withe e decreasing perversity function given by p(Lλ)=2n(n − 1) − 2Aλ for λ a partition of n. For example, when Λ = K ⊃ Qℓ, the category ΛG−umod is semisimple and it follows from the proof of Proposition 2.4 that the functor

Db(KG-umod) −→ Db(KG-umod)

∆λ 7−→ ∆λ[2Aλ − 2n(n − 1)] intertwines (Ei, Fi) and (Ei, Fi). Remark 2.8. We coulde havee deﬁned F and E using the shifted cohomology complex RΓc(Yn,r, Λ)[2 dim Yn,r]. In that case the perverse equivalence should correspond to the decreasing perversity function L(λ) 7−→ −2Aλ.

Another possibility is to use the dual of the latter, which equals RΓ(Yn,r, Λ). In that case the derived equivalence corresponds to the increasing perversity function L(λ) 7−→ 2Aλ, but now it intertwines Fi and Ei with F−i and E−i. Combinatorially, this amounts to changing q to q−1. Note that we can fix this by considering X−1 e e and T −1 instead. Remark 2.9. The natural transformation X comes from a Frobenius endomorphism on an algebraic variety X, which induces an endomorphism of RΓc(X, k) as an object of the bounded derived category. In particular, F cannot be viewed directly as an endomorphism of the complex. When considering eigenspaces of X, we should work with a particular representative of the cohomology complex RΓc(X, k) as defined in [12, Thm. 1.14], which is a bounded complex of finitely generated khF i-modules. Then the functors Fi and Ei correspond to the general- e i ized q -eigenspace of F on RΓc(X, k), and we have decompositions F = Fi and E = E as expected. i e L L 3. Generalizations

We ﬁnish by giving some indications on how to generalize the construction in §2.1 to other induction and restriction functors. Although we do not explain it here, the same constructions should be applicable to other groups, along the lines of the case of Harish-Chandra induction and restriction in unitary groups (see [13]). 14 O. Dudas, M. Varagnolo, E. Vasserot

3.1. Powers of π. In the definition of Yn, one could replace πn/πn−1 by any b power (πn/πn−1) for a fixed b ≥ 1. This yields new functors E and F. However, the endomorphisms X and T can be taken to be the same (up to a possible renormalization). Lemma 2.2 now concerns the action of Ds on the cohomology of the 2b variety X(s ). This was computed when Λ = K ⊃ Qℓ in [8] and can be generalized to the case of integral coefficients whenever q − 1 ∈ Λ×, the only difficult case 2b being when ℓ | q + 1 in which case RΓc(X(s ), O) is homotopy equivalent to

0 −→ Pk −→ Pk −→ PL −→ PL −→ Pk −→ · · · −→ Pk −→ 0 or 0 −→ PL −→ PL −→ Pk −→ Pk −→ PL −→ · · · −→ Pk −→ 0 depending on the parity of b. Here, as in the proof of Lemma 2.2, L is the non- trivial simple module in the principal ℓ-block.

3.2. Square root of π. Using the set of braid reflections Jn = {s2,..., sn−2} one can embed the group Gn−2 in Gn via the Levi subgroup LJn = G1 × Gn−2 × G1. Let wn = s1 ··· sn−1 ··· s1. It centralizes any reflection in Jn and t(wn) is the permutation matrix of the permutation (1,n). Therefore the variety X(Jn, wn) ≃ w t( n)F 2 X(Jn, wn) is endowed with a right action of L ≃ Gn−2 × GL1(q ). We then Jn e define the variety Y to be e n 2 Yn = X(Jn, wn)/GL1(q ).

The induction and restriction functorse Fn,r and En,r obtained from the cohomology of the amalgamated product of these varieties are deﬁned when n − r is an even integer only. The corresponding varieties Yn,r are associated to elements (1,n)(2,n − 1) ··· (m,n − m + 1) where m = (n − r)/2. The endomorphism X is deﬁned as before as a renormalization of the Frobenius, or equivalently of Dwr+2 on Yr+2,r. As for the endomorphism T , it is induced by the braid operator Ds1sr+3 on Yr+4,r so that the relation

(s1sr+3)(s2 ··· sr+2 ··· s2)(s1sr+3)= s1 ··· sr+3 ··· s1

+ in Br+4 yields the relation (c) in Proposition 2.1 between T and X. In this case, for proving relation (b), we are left with computing the action of Ds1sr+3 on the ′ ′ cohomology of the variety XG2×G2 (s1sr+3F ) where F = (1, r + 4)F permutes 2 the two components of G2. This amounts to computing the action of F on the cohomology of the curve XG2 (s), which was achieved by Rouquier in [17]. We deduce 2 (T +1F2 ) ◦ (T − q 1F2 )=0.

In [13] the case of general unitary groups GUn(q) is studied. More precisely, the authors constructed a representation datum (E−, F−,T −,X−) on the category of unipotent representations of ﬁnite unitary groups

ΛGU-umod := ΛGUn(q)-umod Mn≥0 15 coming from Harish-Chandra induction and restriction functors. Numerical evi- dence in [2, §3] suggests the following conjecture. Conjecture 3.1. Let (E−, F−,T −,X−) be the representation datum on ΛGU-umod associated with Harish-Chandra induction and restriction in [13]. Then there exists a derived equivalence Db(ΛGU-umod) ≃ Db(ΛG-umod) which intertwines (E−, F−,T −,X−) and (E, F,T,X). Again, we might hope that this equivalence is perverse, with perversity function given by L(λ) 7−→ n(n − 1) − Aλ. Remark 3.2. The functors F and E studied in this section are called 2-Harish- Chandra induction and restriction functors [2]. For general linear groups one could consider d-Harish-Chandra induction and restriction functors for other integers d using the varieties Xn,d studied in [11]. However the braid group operators which would be needed to deﬁne T are no longer in the positive braid monoid as soon as e d> 1. We still hope that such natural operators exist and satisfy the relation

d (T +1F2 ) ◦ (T − q 1F2 )=0.

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Olivier Dudas, UniversitéParis Diderot, UFR de Mathématiques, Bâtiment Sophie Germain, 5 rue Thomas Mann, 75205 Paris CEDEX 13, France. E-mail: [email protected]

Michela Varagnolo, Laboratoire AGM, D´epartement de Math´ematiques, 2 av. Adolphe Chauvin, 95302 Cergy-Pontoise CEDEX, France. E-mail: [email protected]

Eric´ Vasserot, UniversitéParis Diderot, UFR de Mathématiques, Bâtiment Sophie Germain, 5 rue Thomas Mann, 75205 Paris CEDEX 13, France. E-mail: [email protected]