Categorical actions from Lusztig induction and re- striction on finite general linear groups

striction 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 the- ory.

Introduction

Let G n = GL _n (q) be the general linear group over a finite field with q elements. In order to construct and study the representations of G n with coefficients in a field k of positive characteristic ℓ ∤ q, it is common to consider the chain of subgroups {1} = G 0 ⊂ G 1 ⊂ · · · ⊂ G n ⊂ · · · together with the Harish-Chandra induction and restriction functors − also called parabolic induction and restriction −

kG n - mod

F --

mm

E

kG n+1 - mod .

But there is more to it: Chuang-Rouquier showed in [5] that for any m ≥ 1, the functor F ^m is endowed with a natural action of an affine Hecke algebra of type A m−1 controlling most of the behaviour of these functors. From this action they defined a family of i-induction and i-restriction functors {F i } and { E _i } 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 G n (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 K 0 (C). In addition, these minimal categorifications are unique, as

∗

This research was partially supported by the ANR grant number ANR-13-BS01-0001-01.

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

D ^b (kG n - mod )

F ..

nn

E

D ^b (kG n+1 - mod ).

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( F ² ) satisfing the following relations:

(a) 1 F T ◦ T 1 F ◦ 1 F T = T 1 F ◦ 1 F T ◦ T 1 F , (b) (T + 1 F

) ◦ (T − q1 F

) = 0,

(c) T ◦ (1 F X) ◦ T = qX1 F .

This endows F ^m with a natural action of an affine Hecke algebra. Unlike the case of Harish-Chandra induction and restriction, the natural transformations of F ^m 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 Brou´e-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 {F i } and { E _i }. 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 kG n - umod we need to assume some vanishing properties of the cohomology of Deligne-Lusztig varieties which were conjectured by Brou´e-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 sl b e on

D ^b (kG- umod ) := M

n≥0

D ^b (kG n - umod ).

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.

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 GU n be the finite general unitary group over F q

. There is no Harish-Chandra induction to GU n+1 but there is a Lusztig induction

D ^b (kGU n - mod )

F

..

nn

E

D ^b (kGU n+1 - mod )

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 G n and unipotent characters of GU n which inter- twines Harish-Chandra induction and restriction with this Lusztig induction and restriction as shown in the following diagram:

K 0 (kG n - mod )

∼

[ F ]

..

nn

[E]

K 0 (kG n+1 - mod )

∼

K 0 (kGU n - mod )

[ F

]

..

nn

[ E

]

K 0 (kGU n+1 - mod )

In other words, the g-representations afforded by the Grothendieck groups of unipo- tent representations of general linear and unitary groups are isomorphic. Since both representations come from a categorical g-action, we hope that this might lead the way to a proof that the derived categories D ^b (kG n - mod ) and D ^b (kGU n - 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 F p and F : G −→ G is a Frobenius endomorphism defining an F q -structure on G. Given an F -stable subgroup H of G we will denote by H = H ^F 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 = N ^G (T)/T and the set S ⊂ W of simple reflections. An element s ∈ W lies

in S if and only if B s B / B has dimension 1 in G / B . Since B and T are F-stable,

F induces an automorphism of W which normalizes S.

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 m s,t

denotes the order of st in W then B ⁺ is defined by the presentation B ⁺ = hS | sts | {z } · · ·

m

= tst | {z } · · ·

m

for s, t ∈ Si.

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 = s 1 · · · s r to w = s ₁ · · · s _r where s 1 · · · s r 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 P I the standard parabolic subgroup of G containing B, and by U _I its unipotent radical. The group P _I has a unique Levi complement L _I containing T . The parabolic subgroup of W generated by I, denoted by W I , is the Weyl group of L _I . Let w ∈ W . Assume that w is I-reduced, which means that w has minimal length in the coset W I w. Equivalently, w ⁻¹ 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

X(I, wF ) =

g ∈ G g ⁻¹ F(g) ∈ P _I wP I P _I .

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 ( b F), 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 ⁺ −→

N ^G (T). Then the variety X(I, bF ) has an ´etale covering X(I, e bF ) with Galois group L ^t( _I ^{b )F} . When b = w is the lift of an element w ∈ W then this covering can be defined by

e

X (I, wF ) =

g ∈ G g ⁻¹ F (g) ∈ U _I t( w ) U _I U _I .

The varieties X(I, e bF ) and X(I, bF) are endowed with an action of G by left multiplication. The quotient map X(I, e bF ) ։ X(I, bF ) is equivariant for this action.

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

the field of ℓ-adic integers Q ℓ . 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 D ^b (ΛH - mod ) the corresponding bounded derived category.

Given any quasi-projective algebraic variety X acted on by H, there are well- defined objects RΓ(X, Λ) and RΓ c (X, Λ) in D ^b (ΛH - mod ) representing respectively the ´etale cohomology and the ´etale 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 RΓ c (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 X(I, e bF) yields a complex of (G, L ^t( _I ^{b )F} )-bimodules which is perfect as a complex of G-modules and as a complex of right L ^t( _I ^{b )F} -modules. Indeed, stabilizers under the action of G lie in conjugates of U _I , hence are p-groups, whereas L ^t( _I ^{b )F} acts freely on X(I, e bF ). Using the cohomology complex RΓ c ( X(I, e bF ), Λ) we can form the following pair of adjoint functors

R _{I , b} = RΓ c ( X e ( I , b F ), Λ) ⊗ _{Λ L}

− : D ^b (Λ L ^t( _I ^{b )F} - mod ) −→ D ^b (ΛG- mod ),

∗ R _{I , b} = RHom ΛG (RΓ c ( X e ( I , b F ), Λ), −) : D ^b (ΛG- mod ) −→ D ^b (Λ L ^t( _I ^{b )F} - mod ) called Lusztig induction and 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 [ R I , b ] and [ ^∗ R I , b ] between the Grothendieck groups of Λ L ^t( _I ^{b )F} - mod and ΛG- mod , which were originally con- sidered by Lusztig. Note that by [6, Chap. 6] and [7, Prop. 7.7], these maps depend only on the finite Levi subgroup L ^t( _I ^{b )F} as soon as the Mackey formula holds, which we know for the groups we will encounter by [1]. In particular, [ R I , b ] and [ ^∗ R I , b ] depend only on the image of b in W .

1.4. Action of braid groups. In [3], Brou´e-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 D ^u of X(I, e bF ) which commutes with the actions of G and L ^t( _I ^{b )F} . The particular case of u = b gives D ^b = F, the Frobenius endomorphism.

The crucial property of these endomorphisms is that if b = u ^′ v ^′ = v ^′ F ( u ^′ ) is another decomposition, the endomorphism D ^u D ^u

depends only on the product uu ^′ in B ⁺ . It is denoted by D ^uu

even though uu ^′ is not a prefix of b .

1.5. The central element π . Let w 0 be the longest element in W and w ₀ be

its lift to B ⁺ . Then π = w ² ₀ is a central element in B ⁺ . The ℓ-adic cohomology of

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 A ρ (resp. a ρ ).

Conjecture 1.1 (Brou´e-Michel). The ℓ-adic cohomology of X ( π ) is given by RΓ( X ( π ), Q ℓ ) ≃ M

ρ

ρ ⊗ Hom G (ρ, Q ℓ G/B)[−2A ρ ]

in D ^b (Q _ℓ - mod ), where ρ runs over the set of irreducible unipotent characters. Fur- thermore, the eigenvalue of F on the ρ-isotypic component is q ^a

^+A

.

The constituents of the permutation module Q ℓ G/B are called the principal series representations of G. The conjecture implies that only principal series char- acters 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 F p , and F is the standard Frobenius (a i,j ) 7−→ (a ^q _i,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 group of F q -points T ^F of any split torus T of G is an ℓ ^′ -group.

2.1. The representation datum. Given n ≥ 0, we write G n = GL n (F p ) and G n = G ^F _n = GL n (q), with the convention that G 0 = G 0 = {1}. We will take T n (resp. B n ) to be the diagonal matrices (resp. upper-triangular matrices) in G _n . The corresponding Weyl W n group is isomorphic to the symmetric group.

For i = 1, . . . , n − 1, we denote by s i the permutation matrix corresponding to the transposition (i, i + 1). The braid monoid associated to W n will be denoted by B _n ⁺ . Note that permutation matrices yield a natural F -equivariant Tits homomorphism t : B _n ⁺ −→ N G

(T n ) which factors through W n .

Following [5, §7.3], we view G _n−1 as a subgroup of G _n as follows: we first consider the maximal parabolic subgroup P _n of G _n corresponding to the simple reflections {s 1 , . . . , s n−2 }. It has a unipotent radical U _n and a standard Levi complement which is naturally isomorphic to G _n−1 × G ₁ . If we set V _n = G ₁ ⋉ U _n then we have a decomposition P _n = G _n−1 ⋉ V _n .

For n ≥ 0, we consider the element π _n of the Artin braid group B ⁺ _n of W n

defined by induction by π ₀ = π ₁ = 1 and

π _n = (s 1 · · · s _n−1 s _n−1 · · · s ₁ ) π _n−1

= ( s _n−1 · · · s ₁ s ₁ · · · s _n−1 ) π _n−1 .

The element π _n is the square of the lift of the longest element in W n and thus equals the central element defined in §1.5. Note that the image of π _n in W n is trivial.

Since π _n is central, the element s ₁ · · · s _n−1 s _n−1 · · · s ₁ = π _n / π _n−1 commutes with s _i for i = 1, . . . , n − 2. With I _n = {s 1 , . . . , s _n−2 }, the pair (I n , π _n / π _n−1 ) satisfies the assumptions in §1.2 and one can consider the corresponding parabolic Deligne- Lusztig varieties X(I e n , π _n / π _n−1 ) and X(I n , π _n / π _n−1 ).

The variety X(I e n , π _n / π _n−1 ) is endowed with a left action of G n and a right action of G n−1 × G 1 (note that G 1 ≃ F ^× _q ). Since we will be interested in unipotent representations only we will rather work with the variety

Y _n = X(I e n , π _n / π _n−1 )/G 1 .

It is an ´etale covering of X(I n , π _n / π _n−1 ) with Galois group G n−1 . The definition of Y _n can be made more explicit by considering subvarieties of G _n /V n instead of G _n /U n . More precisely

Y _n ≃

(g, h) ∈ G ² _n | g ⁻¹ h, F (g) ⁻¹ h ∈ V _n (1, . . . , n)V n V ² _n . Given r < n, we form

Y _n,r = Y _n × G

Y _n−1 × G

× · · · × G

Y _r+1 .

It is canonically isomorphic to the quotient of X e ( I _r+1 , π _n / π _r ) by (F ^× _q ) ^n−r . The variety is endowed with a left action of G n and a right action of G r , and therefore RΓ c ( Y _n,r , Λ) ∈ D ^b (ΛG n × G ^opp _r - mod ). In addition, since we assume that q − 1 ∈ Λ ^× , it can be represented by a complex of bimodules which are projective as ΛG n - modules and as ΛG r -modules. Using this complex we can form the following pair of adjoint functors

F _n,r = RΓ c (Y n,r , Λ) ⊗ ΛG

− : D ^b (ΛG r - mod ) −→ D ^b (ΛG n - mod ), E _n,r = RHom ΛG

(RΓ c (Y n,r , Λ), −) : D ^b (ΛG n - mod ) −→ D ^b (ΛG r - mod ).

Note that F _n,r ≃ F _n,n−1 · · · F _r+1,r and E _n,r ≃ E _n,n−1 · · · E _r+1,r . Since these func- tors are defined by bounded complexes of (G n , G r )-bimodules, any endomorphism of the variety Y _n,r which commutes with the left action of G n and the right ac- tion of G r yields an endomorphism of the bimodule RΓ c (Y n,r , Λ) and therefore a natural transformation of F _n,r (and of E _n,r ). Consequently, we can use the braid group operators defined in §1.4 to construct endomorphisms of F _n,r and E _n,r . We will consider two cases.

• The element π _r+1 / π _r obviously commutes with itself, and the correspond- ing operator D π

/π

on Y _r+1,r = Y _r+1 is induced by the Frobenius endo- morphism. We still denote by D π

/π

the endomorphism induced on the functor F _r+1,r and we set X r = q ^−r D π

/ π

.

• The element s r+1 centralizes I r+1 = {s 1 , . . . , s r−1 }, and in particular it com- mutes with π _r . In addition, since π _r+2 is central in B ⁺ _r+2 , it commutes with π _r+2 / π _r . Consequently, there is a corresponding operator D ^s

on Y r+2,r

which induces an isomorphism on the cohomology of Y _r+2,r . We denote by

T r the corresponding endomorphism of F _r+2,r .

Let us define the category

ΛG- mod = M

r≥0

ΛG r - mod .

Then the functors F = L F _r+1,r and E = L E _r+1,r form a pair of adjoint endo- functors of D ^b (ΛG- mod ). They are a particular 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 π _n in W n is trivial. However the functors F and E differ from the Harish-Chandra functors, even when Λ = K ⊃ Q ℓ . Setting X = L

X r

and T = L

T r we obtain endomorphisms X ∈ End( F ) and T ∈ End( F ² ). By adapting the 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( F ² ) satisfy the following relations:

(a) 1 F T ◦ T 1 F ◦ 1 F T = T 1 F ◦ 1 F T ◦ T 1 F , (b) (T + 1 F

) ◦ (T − q1 F

) = 0,

(c) T ◦ (1 F X) ◦ T = qX1 F .

Proof. We start by (a). The endomorphism 1 F T (resp. T 1 F ) of F ³ is induced by the operator D ^s

(resp. D ^s

) on Y r+3,r . Therefore the relation (a) follows directly from the braid relation D ^s

D ^s

D ^s

= D ^s

D ^s

D ^s

(they are both equal to D ^s

s

s

). The same argument, with the equality

s _r+1 ( π _r+1 / π _r ) s _r+1 = s _r+1 ( s ₁ · · · s _r s _r · · · s ₁ ) s _r+1

= s _r+1 (s r · · · s ₁ s ₁ · · · s _r )s r+1

= π _r+2 / π _r+1 in B _r+2 ⁺ yields the relation (c).

The relation (b) was shown to hold for the non-parabolic variety X ( π _r ) in [3, Thm. 2.7] but it can be generalized to our setting. We fix r ≥ 0 and we consider I = I _r = {s 1 , . . . , s _r−1 } and J = I _r ∪ {s r+1 }. We will write for short π = π _r+2 . Then by [7, Prop. 7.19], there is a natural isomorphism of G r+2 × L ^opp _I -varieties

X(J, e π / π J ) × L

X e L

(I, π J / π I ) −→ ^∼ X(I, e π / π I ).

Through this isomorphism, the braid group operator D ^s

on X(I, e π / π _I ) is Id × D ^s

by [7, Prop. 7.24]. Since L _J ≃ G _r × G ₂ and π _J / π _I = s _r+1 s _r+1 , the variety X e L

(I, π _J / π _I ) is isomorphic to G r × X e G

(ss), where s is the unique simple reflection of the Weyl group of G 2 = GL 2 . If we mod out by the torus of G 2 viewed as diag(1, . . . , 1, x, y) ≃ (F ^× _q ) ² then we obtain finally

e

X ( J , π / π _J ) × G

X G

( ss ) −→ ^∼ Y _r+2,r .

In addition, the action of D ^s

on Y _r+2,r corresponds to the action of D ^s on

X G

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

Lemma 2.2. Recall that q(q − 1) ∈ Λ ^× . The braid group operator D ^s on Y ₁ = X G

( ss ) = X ( ss ) induces an endomorphism of RΓ c ( Y ₁ , Λ) satisfying

(D ^s + 1)(D ^s − q) = 0.

Proof. When Λ = K ⊃ Q ℓ , the relation holds by [3, Lem. 2.14]. Here we shall work with Λ = O ⊃ Z ℓ . We first prove that under the assumptions on ℓ, there is no torsion in the cohomology of Y ₁ . Since it is an irreducible variety of dimension 2, the torsion part can only appear in H _c ³ (Y 1 , O). We start by looking at the variety Y ₁ 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) ≃ P 1 , there is a distinguished triangle

RΓ c (Y 1 , O) −→ RΓ c (P 1 , O)[−2](−1) −→ RΓ c (X(s), O)

in D ^b (OG 2 - mod ) which is equivariant for the action of F . Let C be the sum of the generalized eigenspaces of F on RΓ c (Y 1 , O) for the eigenvalues 1 and q ² and let D be the q-generalized eigenspace of F . We get distinguished triangles

C −→ O[−4] −→ H _c ¹ (X(s), O)[−1]

and

D −→ O[−2] −→ O[−2]

in D ^b (OG 2 - 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 kG 2 - stab , which forces the trivial G 2 -module k to be projective. This holds only when ℓ ∤ |G 2 |. Now H ³ (D) = H ³ ( Y ₁ , O) is a torsion module with trivial action of G 2 . This contradicts [8, Prop. 3.3.14] and proves that D ≃ 0. In particular the cohomology of Y ₁ is torsion-free.

To finish the proof it is enough to show that End _D

(OG

- mod ) (C) is torsion-free, since in that case it will embed into the K-algebra End _D

(KG

- mod ) (KC) in which D ^s satifies the relation (D ^s + 1)(D ^s − q) = 0 by [3, Lem. 2.14]. To do so we give two explicit representatives of C as a bounded complex of G 2 -bimodules. We distinguish two cases: if ℓ ∤ q + 1, then ℓ ∤ |G 2 | and

C ≃ H _c ¹ ( X ( s ), O)[−2] ⊕ O[−4].

Here H _c ¹ (X(s), O) is a lattice for the Steinberg character, which is a projective OG 2 -module under the assumptions on ℓ. We deduce that End D

(OG

- 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 G 2 to get

C ≃ 0 −→ P L −→ P L −→ P k −→ 0

where L is the non-trivial simple module in the block, and P L , P k are lattices

lifting the projective covers of the simple modules L and k. One can check that

Hom D

(kG

- mod ) (kC, kC[1]) = 0. Since C is perfect, we can invoke the universal coefficient theorem to conclude that End D

(OG

- mod ) (C) = H ⁰ (REnd OG

(C)) is torsion-free.

Remark 2.3. Note that when ℓ | q+1, the cohomology complex C ≃ RΓ c (X(ss), O) can be obtained as the image of the projective module P k by a perverse equiva- lence 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 algebras. For m ≥ 1, the affine Hecke algebra H ^q _Λ,m is the Λ-algebra generated by T 1 , . . . , T m−1 , X ₁ ^±1 , . . . , X _m ^±1 subject to the relations

• Type A m−1 Hecke relations for T 1 , . . . , T m−1 : (T i + 1)(T i − q) = 0,

T i T i+1 T i = T i+1 T i T i+1 and T i T j = T j T i if |i − j| > 1,

• Laurent polynomial ring relations for X ₁ ^±1 , . . . , X _m ^±1 : X i X j = X j X i and X i X _i ⁻¹ = X _i ⁻¹ X i = 1,

• Mixed relations:

T i X i T i = qX i+1 and X i T j = T j X i if i − j 6= 0, 1.

Then given ( E , F ) a pair of biadjoint functors together with X ∈ End( F ) and T ∈ End( F ² ), the tuple ( E , F , X, T ) is a representation datum if and only if for each m ∈ N, the map

φ F

: H ^q _Λ,m −→ End( F ^m )

X k 7−→ 1 F

X1 F

for all 1 ≤ k ≤ m T l 7−→ 1 F

T 1 F

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 gener-

alized eigenspace of X on F and E . Since X and T satisfy relations of an affine

Hecke algebra, the eigenvalues of X all lie in q ^Z . For i ∈ Z, we define F _i and E _i

to be the q ⁱ -generalized eigenspaces of X on F and E (see Remark 2.9). Then

by definition E = L E _i and F = L F _i . For each i ∈ Z, the functors F _i and E _i

induce linear maps on the Grothendieck group of D ^b (ΛG- mod ) which we denote

by [ F _i ] and [ E _i ]. We would like to show that the linear maps [ E _i ] and [ F _i ] for i ∈ Z

induce an action of a Kac-Moody algebra g on K 0 (ΛG- mod ). This will endow

D ^b (ΛG- mod ) with a structure of categorical g-representation. To this end we will

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 H ^j (X(w), Λ) for some w ∈ W and j ∈ Z.

We denote by ΛG- umod the Serre subcategory of ΛG- mod generated by unipo- tent simple modules. It is a direct summand of ΛG- mod . By [9, 10], the simple unipotent ΛG n -modules are naturally parametrized by partitions of n such that the decomposition map

d : K 0 (KG- umod ) −→ K 0 (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. L λ ) a simple KG n -module (resp. kG n -module) corresponding to λ.

For example, ∆ (n) and L (n) correspond to the trivial representations over K and k respectively, whereas ∆ (1

) is the Steinberg representation. Note that by [14] any irreducible unipotent representation over K (resp. over k) is actually defined over Q ℓ (resp. over F ℓ ), and is absolutely irreducible over this field.

Recall that to a partition λ = (λ 1 ≥ λ 2 ≥ · · · ≥ λ r ) one can associate its Young diagram Y (λ) = {(x, y) | 1 ≤ x ≤ r, 1 ≤ y ≤ λ x }. 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

[ F _i ]([∆ λ ]) =

0 if one cannot add a node of content i to 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( π _r ) with the action of G r and F. A unipotent representation ∆ µ corresponding to a partition µ of r occurs in degree 2r(r−1)−2A µ and with eigenvalue q ^r(r−1)−a

^−A

, with a µ := a ∆

and A µ := A ∆

. Let λ be a partition of n. By [7, Prop. 7.19] and the definition of Y n+1 , we have

Y _n+1 × G

X( π _n ) ≃ X( π _n+1 ) from wich we get, by taking the cohomology

F (∆ λ ) ≃ M

|µ/λ|=1

∆ µ [2n(n − 1) − 2A λ − 2n(n + 1) + 2A µ ]

≃ M

|µ/λ|=1

∆ µ [2(A µ − A λ − 2n)]

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 ( π _n+1 ) by the eigenvalue on ∆ λ in the cohomology of X( π _n ). This equals q ^2n+a

^+A

^−a

^−A

. Now by definition, the action of X coincide with the action of q ⁻ⁿ F, giving, for i ∈ Z

F _i (∆ λ ) ≃ M

k∈ Z

n+a

+A

−a

−A

=i

∆ λ⋆k [2(A λ⋆k − A λ − 2n)]. (1)

Now, for a partition λ = (λ 1 ≥ · · · ≥ λ r ) of n, Lusztig’s a and A-functions are given by

a λ = X

1≤j<l≤r

λ l

and

A λ = n(n − 1)/2 − a λ

= n(n − 1)/2 − X r

l=1

λ l

2

where λ ^∗ denotes the partition conjugate to λ. If µ = λ ⋆ k and (x, y) ∈ Y (µ) is the added node of content k then a µ − a λ = x − 1 and A µ − A λ = n − λ x . Therefore n+ a λ + A λ −a µ −A µ = λ x − x+ 1 = y −x = k. This proves that in the right-hand side of (1), only λ ⋆ i can occur.

By adjunction, we deduce from the proposition that the action of [ E _i ] on unipo- tent characters corresponds to removing nodes of content i. These actions on par- titions are well-known, and they define a structure of sl _Z -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

[ F _i ] d([∆ λ ])

= X

j∈Z

d([∆ λ∗(i+je) ]).

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 unipo-

tent character occurs in even degree cohomology groups of X( π _r ) 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 ( π _r ) vanish.

This shows that K 0 (kG− umod ) has a natural structure of sl b e -module, which is again a Fock space of level 1. As before, this was already known for Harish- 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 E , F e , T , e X e ) be the representation datum associated with Harish-Chandra induction and restriction in [5, §7.3]. There exists a derived self- equivalence of D ^b (ΛG− umod ) which intertwines ( e E , F e , T , e X e ) and ( E , F , T, X).

Furthermore, this equivalence should be perverse, with decreasing perversity function given by p(L λ ) = 2n(n − 1) − 2A λ for λ a partition of n. For example, when Λ = K ⊃ Q ℓ , the category ΛG− umod is semisimple and it follows from the proof of Proposition 2.4 that the functor

D ^b (KG- umod ) −→ D ^b (KG- umod )

∆ λ 7−→ ∆ λ [2A λ − 2n(n − 1)]

intertwines ( E _i , F _i ) and ( E e _i , F e _i ).

Remark 2.8. We could have defined F and E using the shifted cohomology com- plex RΓ c (Y n,r , Λ)[2 dim Y _n,r ]. In that case the perverse equivalence should corre- spond to the decreasing perversity function L(λ) 7−→ −2A λ .

Another possibility is to use the dual of the latter, which equals RΓ(Y n,r , Λ). In that case the derived equivalence corresponds to the increasing perversity function L(λ) 7−→ 2A λ , but now it intertwines F e _i and e E _i 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 and T ⁻¹ instead.

Remark 2.9. The natural transformation X comes from a Frobenius endomor- phism on an algebraic variety X , which induces an endomorphism of RΓ c ( 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 RΓ e c (X, k) as defined in [12, Thm. 1.14], which is a bounded complex of finitely generated khF i-modules. Then the functors F _i and E _i correspond to the general- ized q ⁱ -eigenspace of F on RΓ e c (X, k), and we have decompositions F = L F _i and E = L E _i as expected.

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]).

3.1. Powers of π . In the definition of Y _n , one could replace π _n / π _n−1 by any power ( π _n / π _n−1 ) ^b 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 renor- malization). Lemma 2.2 now concerns the action of D ^s on the cohomology of the variety X(s ^2b ). This was computed when Λ = K ⊃ Q ℓ in [8] and can be general- ized to the case of integral coefficients whenever q − 1 ∈ Λ ^× , the only difficult case being when ℓ | q + 1 in which case RΓ c (X(s ^2b ), O) is homotopy equivalent to

0 −→ P k −→ P k −→ P L −→ P L −→ P k −→ · · · −→ P k −→ 0 or

0 −→ P L −→ P L −→ P k −→ P k −→ P L −→ · · · −→ P k −→ 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 J _n = {s 2 , . . . , s _n−2 } one can embed the group G n−2 in G n via the Levi subgroup L J

= G 1 × G n−2 × G 1 . Let w _n = s ₁ · · · s _n−1 · · · s ₁ . It centralizes any reflection in J _n and t(w n ) is the permutation matrix of the permutation (1, n). Therefore the variety X(J e n , w _n ) ≃ X(J e n , w n ) is endowed with a right action of L ^t( _J ^w

^)F

n

≃ G n−2 × GL ₁ (q ² ). We then define the variety Y _n to be

Y _n = X e (J n , w n )/ GL ₁ (q ² ).

The induction and restriction functors F _n,r and E _n,r obtained from the cohomol- ogy of the amalgamated product of these varieties are defined when n − r is an even integer only. The corresponding varieties Y n,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 D ^w

on Y _r+2,r . As for the endomorphism T , it is induced by the braid operator D ^s

s

on Y _r+4,r so that the relation

(s 1 s r+3 )(s 2 · · · s r+2 · · · s 2 )(s 1 s r+3 ) = s 1 · · · s r+3 · · · s 1

in B _r+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 D ^s

^s

on the cohomology of the variety X G

× G

(s 1 s _r+3 F ^′ ) where F ^′ = (1, r + 4)F permutes the two components of G ₂ . This amounts to computing the action of F ² on the cohomology of the curve X G

(s), which was achieved by Rouquier in [17]. We deduce

(T + 1 F

) ◦ (T − q ² 1 F

) = 0.

In [13] the case of general unitary groups GU n (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 := M

n≥0

ΛGU n (q)- umod

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 D ^b (ΛGU- umod ) ≃ D ^b (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) − A λ .

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 X e n,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 d > 1. We still hope that such natural operators exist and satisfy the relation

(T + 1 F

) ◦ (T − q ^d 1 F

) = 0.

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Olivier Dudas, Universit´e Paris Diderot, UFR de Math´ematiques, Bˆ atiment Sophie Germain, 5 rue Thomas Mann, 75205 Paris CEDEX 13, France.

E-mail: olivier.dudas@imj-prg.fr

Michela Varagnolo, Laboratoire AGM, D´epartement de Math´ematiques, 2 av. Adolphe Chauvin, 95302 Cergy-Pontoise CEDEX, France.

E-mail: michela.varagnolo@u-cergy.fr

Eric Vasserot, Universit´e Paris Diderot, UFR de Math´ematiques, Bˆ ´ atiment Sophie Germain, 5 rue Thomas Mann, 75205 Paris CEDEX 13, France.

E-mail: eric.vasserot@imj-prg.fr