On the endomorphism algebra of modular Gelfand-Graev representations

HAL Id: hal-00168266

https://hal.archives-ouvertes.fr/hal-00168266v1

Preprint submitted on 27 Aug 2007 (v1), last revised 24 Apr 2008 (v2)

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

On the endomorphism algebra of modular Gelfand-Graev representations

Cédric Bonnafé, Radha Kessar

To cite this version:

Cédric Bonnafé, Radha Kessar. On the endomorphism algebra of modular Gelfand-Graev representa- tions. 2007. �hal-00168266v1�

hal-00168266, version 1 - 27 Aug 2007

GELFAND-GRAEV REPRESENTATIONS

C´EDRIC BONNAF´E¹AND RADHA KESSAR²

This paper is dedicated to Toshiaki Shoji, on his sixtieth birthday.

Abstract. We study the endomorphism algebras of a modular Gelfand-Graev representation of a finite reductive group by investigating modular properties of homomorphisms constructed by Curtis and Curtis-Shoji.

Let G be a connected reductive group defined over an algebraic closure F of the field ofp-elementsF_p and suppose that it is endowed with a Frobenius endomorphism F : G → G relative to an F_q-structure. Since the work of Lusztig, it has been natural to ask to what extent the theory of the representations of G^F depends on q. For example, it was shown by Lusztig that the unipotent characters of G^F are parametrized by a set which is independent ofq(the set depends solely on the Weyl group of G and on the action ofF on this Weyl group).

On the side of ℓ-modular representations (where ℓ is a prime different from p), the work of Fong and Srinivasan on the general linear and unitary groups [FS1] and on the classical groups [FS2], then that of Brou´e, Malle and Michel (introducing the notion ofgeneric groups[BMM]), give evidence of analogous reults. For instance, in the split case the unipotent ℓ-blocks of G^F only depend on the order of q moduloℓ (and not on the value of q itself).

Let (K,O, k) denote anℓ-modular system, sufficiently large. In this article we will study the endomorphism algebra H^G_(d) of a modular Gelfand-Graev representation Γ^G_(d)ofG^Fd(this is a projectiveOG^Fd-module). We will study also the corresponding unipotent parts b^G_(d)H^G_(d) and b^G_(d)Γ^G_(d) (here, b^G_(d) denotes the sum of the unipotent blocks ofOG^Fd). We make the following conjecture, which is related to the question mentioned above:

Conjecture 1. If ℓ does not divide [G^Fd : G^F], then the O-algebras b^G_(d)H^G_(d) and b^G₍₁₎H^G₍₁₎ are isomorphic.

Date: August 27, 2007.

1991Mathematics Subject Classification. According to the 2000 classification: Primary 20C20;

Secondary 20G40.

1

The OG^Fd-module b^G_(d)Γ^G_(d) is projective and indecomposable : it is the projective cover of the modular Steinberg module. Conjecture 1, if proven, would show that the endomorphism algebra of this module does not depend too much onq.

In this article, we approach Conjecture 1 by the study of a morphism _KCur^G_L,(d) : KH^G_(d) →KH^L_(d) (whereL is anF^d-stable Levi subgroup of a parabolic subgroup of G). WhenT is a maximalF^d-stable torus of G, this morphism was constructed by Curtis [C, Theorem 4.2] and it is defined overO (i.e. there exists a morphism of al- gebras Cur^G_T,(d) :H^G_(d) → H^T_(d)=OT^Fd such that_KCur^G_T,(d) is obtained from Cur^G_T,(d) by extension of scalars). We will also consider a product of Curtis homomorphisms

Cur^G_(d) :H^G_(d)−→ Y

T∈T^{F d}

OT^Fd,

where T is the variety of maximal tori of G. Finally, we will study a morphism of K-algebras _K∆^G : KH^G_(d) → KH^G₍₁₎ defined by Curtis and Shoji [CS, Theorem 1].

With this notation, we can state Conjecture 1 more precisely : Conjecture 2. With the notaion above, we have:

(a) _KCur^G_L is defined over O. (b) _K∆^G is defined over O.

(c) If ℓ does not divide [G^Fd : G^F], then _K∆^G induces an isomor- phism b^G_(d)H_(d)^G ≃b^G₍₁₎H^G₍₁₎.

The main results of this article are obtained under the hypothesis thatℓdoes not divide the order of the Weyl group W of G.

Theorem. If ℓ does not divide |W|, then Conjecture 2 holds.

Statement (a) is proved in Corollary 3.12 ; statement (b) in Theorem 4.4 ; state- ment (c) is shown Theorem 4.9. In order to obtain our theorem, we proved two more precise results which do not necessarily hold when ℓ does divide |W|.

Theorem 3.7. If ℓ does not divide |W|, then

Im(Cur^G_(d)) = Im(_KCur^G_(d))∩ Y

T∈T^{F d}

OT^Fd.

Theeorem 3.13. If ℓ does not divide |W|, then b^G_(d)H^G_(d) ≃(OS)^{NGF d(S)}, where S is a Sylowℓ-subgroup of G^Fd.

This article is organized as follows. In the first section, we recall the definitions of the Gelfand-Graev representations as well as some of the principal properties of their endomorphism algebras (commutativity for example). In the second section, we construct the generalisation of the Curtis homomorphism. In the third part we study the product of Curtis homomorphisms and prove, amongst other things, Theorems 3.7 and 3.13 stated above. In the last part, we study the Curtis-Shoji homomorphsim and prove statement (c) of Conjecture 2 whenℓdoes not divide|W|.

Notation - If A is a finite dimensional algebra over a field, we denote by R(A) the Grothendieck group of the category of finitely generated A-modules. If M is a finitely generated A-module, we denote by [M] its class in R(A). The opposite algebra ofA will be denoted byA^◦.

All along this paper, we fix a prime number p, an algebraic closure Fof the finite field withpelementsF_p, a prime numberℓdifferent frompand an algebraic extension K of the ℓ-adic field Q_ℓ. Let O be the ring of integers of K, let l be the maximal ideal of O and let k denote the residue field of O: k is an algebraic extension of the finite field F_ℓ. Throughout this paper, we assume that the ℓ-modular system (K,O, k) is sufficiently large for all the finite groups considered in this paper.

If Λ is a commutative O-algebra (for instance Λ = k or K), and if M is an O- module, we set ΛM = Λ⊗O M. If f : M → N is a morphism of O-modules, we define Λf : ΛM → ΛN to be the morphism IdΛ⊗Of. If V is a free left Λ-module, we denote by V^∗ = HomΛ(V,Λ) its dual: if V is a left A-module for some Λ-algebra A, then V^∗ is seen as a rightA-module.

IfGis a finite group, we denote by IrrGthe set of irreducible characters ofGover K. If χ ∈ IrrG, let eχ (or e^G_χ if we need to emphasize the ambient group) denote the associated central primitive idempotent of KG :

eχ = χ(1)

|G|

X

g∈G

χ(g⁻¹)g.

The conjugacy relation in G is denoted by∼ or∼G if necessary.

1. Background material

1.A. The set-up. We fix once and for all a connected reductive algebraic groupG over F and we assume that it is endowed with an isogeny F : G → G such that some power of F is a Frobenius endomorphism of G with respect to some rational structure onGover a finite extension ofF_p. We denote byqthe positive real number

such that, for every δ>1 such that F^δ is a Frobenius endomorphism of G over a finite field with r elements, we have r=q^δ.

1.B. Gelfand-Graev representations. We fix anF-stable Borel subgroupBG of G and an F-stable maximal torus of BG. Let UG denote the unipotent radical of BG. We fix once and for all a regular linear character ψ :U^F_G→ O^× ⊂K^× (in the sense of [DLM1, Definition 2.3]). Since|U^F_G|=q^dimUG is a power ofp, the primitive central idempotent e_ψ of KU^F_G belongs to OU^F_G. We denote byO_ψ the projective OU^F_G-module OU^F_Ge_ψ: it is O-free of rank one and is acted on by U^F_G through ψ.

Let

Γ^G=OG^Feψ ≃Ind^G_U^FF GOψ.

Then Γ^G is a projective OG^F-module; the corresponding representation is called a Gelfand-Graev representation of G^F.

LetH^G denote the endomorphism algebra of the OG^F-module Γ^G. We have (1.1) H^G≃(eψOG^Feψ)^◦

Since OG^F is a symmetric algebra, we have that

(1.2) H^G is symmetric.

The next result is much more difficult (see [S, Theorem 15] for the general case):

Theorem 1.3. The algebra H^G is commutative.

Therefore, we shall identify the algebras H^G and eψOG^Feψ.

If Λ is a commutative O-algebra, we denote by e^Λ_ψ the idempotent 1Λ⊗O eψ of ΛU^F_G= Λ⊗OOU^F_G. Since Γ^Gis projective, the ΛG^F-module ΛΓ^Gis also projective and its endomorphism algebra is ΛH^G(since it is isomorphic to HomΛ(ΛΓ^G,Λ)⊗_ΛG^F ΛΓ^G). We have of course (taking into account that H^G is symmetric)

(1.4) ΛH^G=e^Λ_ψΛG^Fe^Λ_ψ. Since KG^F is split semisimple,

(1.5) The algebra KH^G is split semisimple.

Remark 1.6 - There might be several Gelfand-Graev representations of OG^F. But they are all conjugate by elements g ∈Gsuchg⁻¹F(g) belongs to the centre of G, and this gives a parametrization of Gelfand-Graev representations by the group of F-conjugacy classes in the centre of G(see [DLM1, 2.4.10]). In particular, their endomorphism algebras are all isomorphic.

Moreover, if the centre of G is connected, there is only one (up to isomorphism) Gelfand-Graev representation. In special orthogonal or symplectic groups in odd

characteristic, there are two (isomorphism classes of) Gelfand-Graev representa- tions.

1.C. Representations of KH^G. Let (G^∗, F^∗) be a dual pair to (G, F) in the sense of Deligne and Lusztig [DL, Definition 5.21]. We denote by G^∗_sem the set of semisimple elements ofG^∗. If s∈G^∗F_sem^∗, we denote by (s)_G^∗F∗ its conjugacy class in G^∗F∗ and by E(G^F,(s)_G^∗F∗) the associated rational Lusztig series (see [DM, Page 136]). We denote byχ^G_s the unique element ofE(G^F,(s)_G^∗F∗) which is an irreducible component of the character afforded byKΓ^G. We view it as a functionKG^F →K and we denote by χ^H_s^G its restriction to KH^G. Then (see [DL, Theorem 10.7] for the case where the centre of G is connected and [A] for the general case; see also [B3, Remark of Page 80] for the case where F is not a Frobenius endomorphism),

(1.7) [KΓ^G] = X

(s)G∗F∗∈G^∗F_sem^∗/∼

χ^G_s .

Therefore, the next proposition is a particular case of [CR, Theorem 11.25 and Corollaries 11.26 and 11.27], taking into account thatKH^Gis semisimple and com- mutative (or, equivalently, that KΓ^G is multiplicity free):

Proposition 1.8. We have:

(a) The map s 7→ χ^H_s^G induces a bijection between the set of G^∗F∗-conjugacy classes of semisimple elements ofG^∗F∗ and the set of irreducible characters of KH^G.

(b) The map s 7→ e_χ^G_s eψ induces a bijection between the set of G^∗F∗-conjugacy classes of semisimple elements ofG^∗F∗ and the set of primitive idempotents of KH^G.

Moreover, if s∈G^∗F_sem^∗, then:

(c) We have χ^H_s^G(e_χ^G_s e_ψ) = 1 and χ^H_s^G(e_χ^G

t e_ψ) = 0 if t∈G^∗F_sem^∗ is not conjugate to s in G^∗F∗.

(d) e_χ^G_s e_ψ is the primitive idempotent of KH^G associated with the irreducible character χ^H_s^G.

(e) TheKG^F-module KG^Fe_χ^G_s eψ is irreducible and affords the character χ^G_s . (f) If χ∈ E(G^F,(s)_G^∗F∗) and if χ6=χ^G_s , then χ(h) = 0 for all h∈KH^G. Since KH^G is split and commutative, all its irreducible representations have di- mension one. In other words, all its irreducible characters are morphisms of K- algebras KH^G → K. So, as a consequence of the Proposition 1.8, we get that the

map (1.9)

χ^HG : KH^G −→ Y

(s)_G∗F∗∈G^∗_sem^F∗/∼

K h 7−→ (χ^H_s^G(h))_(s)

G∗F∗∈G^∗_sem^F∗/∼

is an isomorphism of K-algebras. It corresponds to the decomposition (1.10) KH^G= ⊕

(s)_G∗F^∗∈G^∗_sem^F∗/∼

KH^Ge_χ^G_s eψ.

2. A generalization of the Curtis homomorphisms

In [C, Theorem 4.2], Curtis constructed a homomorphism of algebras fT :H^G→ OT^F, for T an F-stable maximal torus of G (in fact, Curtis constructed a homo- morphism of algebrasKH^G→KT^F but it is readily checked from his formulas that it is defined over O). We propose here a generalization of this construction to the case where T is replaced by an F-stable Levi subgroup L of a parabolic subgroup of G: we then get a morphism KH^G → KH^L (note that, if L is a maximal torus, then H^L = OT^F). We conjecture that this morphism is defined over O and prove it whenever G⁺(G,L,P) holds or whenever Lis a maximal torus (see Theorem 2.7) or whenever ℓ does not divide the order of W (see Corollary 3.12).

2.A. A morphism KH^G →KH^L. Let P be a parabolic subgroup of G and assume that Padmits an F-stable Levi complement L.

The Gelfand-Graev representation Γ^G of OG^F having been fixed, there is a well- defined (up to isomorphism) Gelfand-Graev representation Γ^L ofOL^F associated to it [B3, Page 77] (see also [B1]). We fix anF-stable Borel subgroup BL of L and we denote byULits unipotent radical. We fix once and for all a regular linear character ψL of U^F_L such that Γ^L= Ind^L_U^FF

L OψL =OL^FeψL. We identifyH^L with eψLOL^FeψL. We also fix an F^∗-stable Levi subgroup L^∗ of a parabolic subgroup ofG^∗ dual to L (this is well-defined up to conjugacy by an element of G^∗F∗: see [DM, Page 113]).

We then define _KCur^G_L KH^G → KH^L as the unique linear map such that, for any semisimple element s∈G^∗F∗,

KCur^G_L(e_χ^G_s eψ) = X

(t)_L∗F^∗∈L^∗F_sem^∗/∼

t∈(s)_G∗F∗

e_χ^L_teψL.

Note that this does not depend on the choice of the representativesin its conjugacy class.

Proposition 2.1. The map _KCur^G_L is an homomorphism of algebras. Moreover, if s∈L^∗F_sem^∗, then

χ^H_s^L ◦_KCur^G_L =χ^H_s^G.

Proof. Since the image of an idempotent is an idempotent (and sinceKH^GandKH^L are split semisimple and commutative), we get the first statement. The second is obtained by applying both sides to each primitive idempotent e_χ^G_t eψ of KH^G

(t∈G^∗F_sem^∗).

Another easy consequence of the definition is the following

Proposition 2.2. If M is an F-stable Levi subgroup of a parabolic subgroup of G and if L⊂M, then _KCur^M_L ◦_KCur^G_M= _KCur^G_L.

2.B. Deligne-Lusztig functors and Gelfand-Graev representations. Let P be a parabolic subgroup of G and assume that P admits an F-stable Levi comple- ment L. Let V denote the unipotent radical ofP. We set

Y^G_P ={gV ∈G/V |g⁻¹F(g)∈V·F(V)}

and dP = dim(V)−dim(V∩F(V)). Then Y_P^G is a locally closed smooth variety of pure dimension dP. If Λ = O, K or O/lⁿ, the complex of cohomology with compact support of Y_P^G with coefficients in Λ, which is denoted by RΓc(Y_P^G,Λ), is a bounded complex of (ΛG^F,ΛL^F)-bimodules which is perfect as a complex of left ΛG^F-modules and is also perfect as a complex of right ΛL^F-modules (see [DL,§3.8]).

Its i-th cohomology group is denoted byH_cⁱ(Y^G_P,Λ): it is a (ΛG^F,ΛL^F)-bimodule.

For Λ =O, this complex induces two functors between bounded derived categories R^G_L⊂P : D^b(OL^F) −→ D^b(OG^F)

C 7−→ RΓ_c(Y^G_P,O)⊗_OL^F C and

∗R^G_L⊂P: D^b(OG^F) −→ D^b(OL^F)

C 7−→ R Hom^•_OGF(RΓ_c(Y^G_P,O), C).

These functors are respectively calledDeligne-Lusztig inductionand restriction. By extending the scalars to K, they induce linear maps between the Grothendieck groups R^G_L⊂P:R(KL^F)→ R(KG^F) and ^∗R^G_L⊂P:R(KG^F)→ R(KL^F). We have

R^G_L⊂P[M] = X

i>0

(−1)ⁱ[H_cⁱ(Y_P^G, K)⊗KL^F M]

and ^∗R^G_L⊂P[N] = X

i>0

(−1)ⁱ[H_cⁱ(Y_P^G, K)^∗⊗_KG^F N] for allKG^F-modules N and all KL^F-modulesM.

If (g, l)∈KG^F ×KL^F, we set Tr^G_L⊂P(g, l) = X

i>0

(−1)ⁱTr((g, l), H_cⁱ(Y_P^G, K)).

If (g, l) ∈ G^F ×L^F, then Tr^G_L⊂P(g, l) is a rational integer which does not depend on the prime number ℓ (see [DL, Proposition 3.3]). IfχM (respectively χN) denotes the character afforded by a KL^F-module M (respectively a KG^F-moduleN), then the character afforded by the virtual moduleR^G_L⊂P[M] (respectively ^∗R_L⊂P^G [N]) will be denoted by R^G_L⊂PχM (respectively ^∗R_L⊂P^G χN): it satisfies

R^G_L⊂PχM(g) = 1

|L^F| X

l∈L^F

Tr^G_L⊂P(g, l)χM(l⁻¹)

(respectively ^∗R_L⊂P^G χ_N(l) = 1

|G^F| X

g∈G^F

Tr^G_L⊂P(g, l)χ_N(g⁻¹) ) for allg ∈G^F (respectively l ∈L^F).

Comments (independence on the parabolic) - If P^′ is another parabolic subgroup ofGhavingLas a Levi complement, then the Deligne-Lusztig varietiesY_P^G and Y_P^G′ are in general non-isomorphic: they might even have different dimension (however, note that (−1)^dP = (−1)^dP′, i.e. dP ≡ dP^′ mod 2). As a consequence, the Deligne-Lusztig functorsR^G_L⊂P andR^G_L⊂P^′ can be really different. However, it is conjectured in general thatR^G_L⊂P =R^G_L⊂P^′ and ^∗R^G_L⊂P= ^∗R_L⊂P^{G ′}. This is equivalent to say that Tr^G_L⊂P= Tr^G_L⊂P^′.

For instance, we have Tr^G_L⊂P = Tr^G_L⊂P^′ if L is a maximal torus [DL, Corollary 4.3], or if P and P^′ are F-stable (this is due to Deligne: a proof can be found in [DM, Theorem 5.1]), or ifF is a Frobenius endomorphism and q6= 2 (see [BM]). In all these cases, this fact is a consequence of the Mackey formula for Deligne-Lusztig maps.

The Gelfand-Graev representation Γ^L satisfies the following property:

Theorem 2.3. Assume that one of the following three conditions is satisfied:

(1) P isF-stable.

(2) The centre of L is connected.

(3) p is almost good for G, F is a Frobenius endomorphism of G and q is large enough.

Then ^∗R^G_L⊂P[KΓ^G] = (−1)^dP[KΓ^L].

Proof. (1) is due to Rodier: a proof may be found in [DLM1, Theorem 2.9]. (2) is proved in [DLM1, Proposition 5.4]. For (3) see [DLM2, Theorem 3.7], [B2, Theorem

15.2] and [B3, Theorem 14.11].

It is conjectured that the above theorem holds without any restriction (onp,q,F or the centre ofL...). However, at the time of the writing of this paper, this general conjecture is still unproved. So we will denote by G(G,L,P) the property

(G(G,L,P)) ^∗R^G_L⊂P[KΓ^G] = (−1)^dP[KΓ^L].

Most of the results of this subsection will be valid only under the hypothesis that G(G,L,P) holds. In light of the above theorem shows, and as there are many other indications that G(G,L,P) holds in general, this should not be viewed as a big restriction.

In fact, there is also strong evidence that the perfect complex of OL^F-modules

∗R^G_L⊂PΓ^Gis concentrated in degree dP: more precisely, it is conjectured [BR2, Con- jecture 2.3] that

(G⁺(G,L,P)) ^∗R^G_L⊂PΓ^G≃Γ^L[−dP]

The conjectural property G⁺(G,L,P) is a far reaching extension of G(G,L,P). It is known to hold only ifP is F-stable (see Theorem 2.3 (1) or [BR2, Theorem 2.1]

for a module-theoretic proof) or if L is a maximal torus and (P, F(P)) lies in the orbit associated with an element of the Weyl group which is a product of simple reflections lying in different F-orbits [BR2, Theorem 3.10]. Of course, a proof of this conjecture would produce immediately a morphism of O-algebras H^G → H^L (which is uniquely determined since H^L is commutative). However, as we shall see in this section, we only need that G(G,L,P) holds to get the following result:

Proposition 2.4. If G(G,L,P) holds, then, for all h∈KH^G⊂KG^F,

KCur^G_L(h) = (−1)^dP X

s∈L^∗_sem^F∗/∼_L∗F^∗

Tr^G_L⊂P(h, e_χ^L_seψL)e_χ^L_seψL.

Proof. We assume throughout this proof that G(G,L,P) holds. We denote by Γ^G the character afforded by the module KΓ^G. Let f : KH^G → KH^L be the map defined by

f(h) = (−1)^dP X

s∈L^∗F_sem^∗/∼L∗F∗

Tr^G_L⊂P(h, e_χ^L_seψL)e_χ^L_seψL.

Lets ∈L^∗F_sem^∗. In order to prove the proposition, we only need to check that (?) χ^H_s^L ◦f =χ^H_s^L ◦_KCur^G_L .

First, note that hχ^L_s,Γ^Li_L^F = 1 so, by adjunction, and since G(G,L,P) holds, we have hR^G_L⊂Pχ^L_s,Γ^Gi_G^F = (−1)^dP. Sinceχ^G_s is the unique irreducible constituent of Γ^G lying in E(G^F,(s)_Γ^∗F∗) and since all the irreducible constituents of R^G_L⊂Pχ^L_s

belong to E(G^F,(s)_G^∗F∗) (see for instance [B3, Theorem 11.10]), we have R^G_L⊂Pχ^L_s = (−1)^dPχ^G_s + X

χ∈E(G^F,(s)_G∗F∗) χ6=χ^G_s

m_χχ

for some mχ ∈Z. By Proposition 1.8 (f), we have (∗) R^G_L⊂Pχ^L_s

(h) = (−1)^dPχ^G_s (h) = (−1)^dPχ^H_s^G(h) =χ^H_s^L(_KCur^G_L(h)) for allh∈KH^G. On the other hand, we have

χ^H_s^L(f(h)) = (−1)^dPTr^G_L⊂P(h, e_χ^L_seψL).

But, since the actions of h and of e_χ^L_seψL on the cohomology groups H_cⁱ(YP, K) commute and since e_χ^L_seψL is an idempotent, we have that Tr^G_L⊂P(h, e_χ^L_seψL) is the trace of h on the virtual module

X

i>0

(−1)ⁱ[H_cⁱ(YP, K)e_χ^L_seψL)] = X

i>0

(−1)ⁱ[H_cⁱ(YP, K)⊗_KL^F KL^Fe_χ^L_seψL)].

Now, by Proposition 1.8 (e), theKL^F-moduleKL^Fe_χ^L_seψL affords the characterχ^L_s. So it follows that

(∗∗) Tr^G_L⊂P(h, e_χ^L_se_ψL) = R^G_L⊂Pχ^L_s (h).

So, (?) follows from the comparison of (∗) and (∗∗).

Proposition 2.5. If B is a Borel subgroup ofG and if T is a maximal torus of B, then G(G,T,B)holds and _KCur^G_T coincides with Curtis homomorphism f_T defined in [C, Theorem 4.2]. We have, for all h ∈KH^G,

KCur^G_T(h) = 1

|T^F| X

t∈T^F

Tr^G_T⊂B(h, t)t⁻¹.

Remark - The formula given in Proposition 2.5 gives a concise form for Curtis homomorphism. It can be checked directly, using the character formula [DM, Propo- sition 12.2], that this indeed coincides with the formulas given by Curtis in terms of Green functions [C, 4.3]. However, we shall give a more theoretical proof of this coincidence.

Proof. Since the centre of T is connected, G(G,T,B) holds by Theorem 2.3 (2).

Also,U_T = 1,ψ_T = 1, so KH^T =KT^F. So the primitive idempotents ofKH^T are the primitive idempotents ofKT^F and the formula given above can be obtained by a straightforward computation.

Now, let T^∗ be an F^∗-stable maximal torus of G^∗ dual to T. If s ∈ T^∗F∗, then χ^T_s = χ^H_s^T and, by [C, Theorem 4.2], Curtis homomorphism fT : KH^G → KT^F satisfies

χ^T_s ◦fT =χ^H_s^G.

Since χ^HT is an isomorphism of K-algebras, we get from Proposition 2.1 that fT =

Cur^G_T.

Remark 2.6 - Ifχis a class function onL^F (which can be seen as a class function onKL^F) and if G(G,L,P) holds, then we have

χ(Cur^G_L(h)) = (−1)^dPR^G_L⊂P(χ)(h)

for all h∈ KH^G. For this, one may assume that χ∈ IrrL^F. Let s ∈L^∗F_sem^∗ be such that χ∈ E(L^F,(s)_L^∗F∗). If χ=χ^L_s, then this is the equality (∗) in the proof of the Proposition 2.1. If χ 6= χ^L_s, we must show that R_L⊂P^G (χ)(h) = 0 for all h ∈ KH^G (see Proposition 1.8 (f)). Let γ ∈ IrrG^F be such that hγ, R_L⊂P^G χi_G^F 6= 0. Then γ ∈ E(G^F(s)_G^∗F∗) (see for instance [B3, Theorem 11.10]) and, by Proposition 1.8 (f), it is sufficient to show that γ 6=χ^G_s . But

hχ^G_s , R^G_L⊂Pχi_G^F =hΓ^G, R^G_L⊂Pχi_G^F =hΓ^L, χi_L^F = 0.

This shows the result.

2.C. A morphism H^G → H^L. We conjecture that, in general, _KCur^G_L(H^G) ⊂ H^L. At this stage of the paper, we are only able to prove it in the following cases (in Corollary 3.12, we shall see that this property also holds inℓdoes not divide the order of W):

Theorem 2.7. We have:

(a) If G⁺(G,L,P) holds, then _KCur^G_L(H^G) ⊂ H^L and the resulting morphism of O-algebraH^G→ H^L coincides with the functorial morphism coming from the isomorphism ^∗R^G_L⊂PΓ^G≃Γ^L[−dP].

(b) If L is a maximal torus, then _KCur^G_L(H^G)⊂ H^L.

Proof. (b) follows easily from Proposition 2.1 and from the well-known fact that, if (g, l)∈G^F ×L^F, then |L^F| divides Tr^G_L⊂P(g, l) because L is a maximal torus.

(a) The complex RΓc(Y_P^G,O) is perfect as a complex of left OG^F-modules.

Therefore, we have ^∗R^G_L⊂PC = RΓc(Y^G_P,O)^∗⊗_OG^F C for any complex C of OG^F- modules. If G⁺(G,L,P) holds, then this means that we have an isomorphism RΓc(Y_P^G,O)^∗eψ ≃ Γ^L[−dP]. In particular, the complex RΓc(Y_P^G,O)^∗eψ is concen- trated in degree dP. Therefore, there exists an (OL^F,H^G)-bimodule P such that RΓc(Y_P^G,O)^∗eψ ≃ P[−dP]. Moreover, as a left OL^F-module, we have an isomor- phism α:OL^Fe_ψL

−→∼ P. This induces a morphism

˜

α: H^G −→ H^L h 7−→ α⁻¹(α(e_ψL)h).

The morphism ˜α : H^G → H^L does not depend on the choice of α because H^L is commutative. This morphism can be extended to a morphism Kα˜:KH^G→KH^L, h7→ Kα⁻¹(Kα(eψL)h). Now the Theorem would follow if we show thatKα˜= _KCur^G_L. So let s ∈ G^∗F_sem^∗. Let E be a set of representatives of L^F-conjugacy classes which are contained in L^∗F∗ ∩(s)_G^∗F∗ and let e =P

t∈EeψLe_χ^L_t. Then ^∗R^G_L⊂Pχ^G_s = (−1)^dPP

t∈Eχ^L_t. In particular,α induces an isomorphism KP e_χ^L_s ≃KL^Fe.

So this shows that Kα(e˜ ψe_χ^G_s ) = e, as desired.

2.D. Truncation at unipotent blocks. We denote byb^Gthe sum of theunipotent block idempotents of G^F. In other words,

b^G= X

s∈G^∗_sem^F∗/∼

sis anℓ-element

X

χ∈E(G^F,(s)_G∗F^∗)

e_χ.

The algebra H^G is a module over the centre of theO-algebraOG^F: so b^GH^G is an O-algebra with unitb^Ge_ψ. Note that

b^Geψ = X

s∈G^∗_sem^F∗/∼

sis anℓ-element

e_χ^G_s eψ.

Now, by definition, we get

KCur^G_L(b^Geψ) = b^LeψL. In particular,

(2.8) _KCur^G_L(b^GKH^G)⊂b^LKH^L.

Let us also recall for future reference the following classical fact:

Proposition 2.9. The projective OG^F-moduleb^GΓ^G is indecomposable.

Proof. For the sake of completeness, we recall a proof of this proposition. Assume thatb^GΓ^G=P1⊕P2 wherePi are two non-zero sub-OG^F-modules. ThenP1 andP2

are projective. Since they belong to unipotent blocks, the characters of the KG^F- modules KP1 and KP2 must both contain a unipotent character (see for instance [CE, Theorem 9.10]). But this is impossible since the character of KΓ^G contains only one unipotent character (namely, the Steinberg character).

Corollary 2.10. The algebra b^GH^G is local.

3. Glueing Curtis homomorphisms for maximal tori

If B is a Borel subgroup of G and if T is an F-stable maximal torus of B, we then write R_T^G, and Tr^G_T for the maps R^G_T⊂B and Tr^G_T⊂B (see the comments at the end of subsection 2.B and Proposition 2.1 (c)).

LetTGdenote anF-stable maximal torus ofBG. We setW =NG(TG)/TG. For each w ∈ W, we fix an element g ∈ G such that g⁻¹F(g) belongs to NG(TG) and represents w. We then set Tw = gTGg⁻¹. We then define, following [CS, Lemma 1],

Cur^G: H^G −→ Y

w∈W

OT^F_w h 7−→ Cur^G_Tw(h)

w∈W. The aim of this section is to study the map Cur^G.

3.A. Properties of _KCur^G. Before studying Cur^G, we shall study the simpler map

KCur^G. It turns out that _KCur^G is injective and it is relatively easy to describe its image: both facts were obtained by Curtis and Shoji [CS, Lemmas 1 and 5] but we shall present here a concise proof.

We first need to introduce some notation. Ifw∈W, we fix anF^∗-stable maximal torus T^∗_w dual to Tw. If t ∈ T^∗F_w ^∗, then χ^t_t^w is a linear character of T^F_n. If s is a semisimple element of G^∗F∗, we set

e^G(s) = X

t∈(s)G∗F∗∩T^∗F_w ^∗

e_χ^Tw t

w∈W ∈ Y

w∈W

KT^F_w. Then, by definition, we have

(3.1) _KCur^G(e_χ^G_s eψ) =e^G(s).

Since (e^G(s))_(s)∈G^∗F_sem^∗_/∼ is a K-linearly independent family in Q

w∈WKT^F_w, we get:

Proposition 3.2 (Curtis-Shoji). The map _KCur^G is injective and Im_KCur^G= ⊕

(s)∈G^∗F_sem^∗/∼

Ke^G(s).

Corollary 3.3. The map Cur^G is injective.

We shall now recall a characterization of elements of the image of _KCur^G which was obtained by Curtis and Shoji [CS, Lemma 5]. We need some notation. LetS^G denote the set of pairs (w, θ) such that w ∈ W and θ is a linear character of T^F_w (which may also be viewed as a morphism of algebras OT^F_w → O or KT^F_w → K).

If (w, θ) and (w^′, θ^′) are two elements of S^G, we write (w, θ) ≡ (w^′, θ^′) if (T_w, θ) and (Tw^′, θ^′) lie in the same rational series (see for instance [B3, Definition 9.4] for a definition).

Corollary 3.4(Curtis-Shoji). Lett= (tw)w∈W ∈Q

w∈W KT^F_w. Thent∈Im_KCur^G if and only if, for all (w, θ), (w^′, θ^′) ∈ S^G such that (w, θ) ≡ (w^′, θ^′), we have θ(tw) =θ^′(tw^′).

3.B. Symmetrizing form. The O-algebraH^G is symmetric. In particular, theO- algebra Im Cur^G is symmetric (see Corollary 3.3). We shall give in this subsection a precise formula for the symmetrizing form on Im Cur^G. For this, we introduce the following symmetrizing form

˜

τ : Y

w∈W

KT^F_w −→ K

(xw)w∈W 7−→ 1

|W| X

w∈W

τw(xw), where τw :KT^F_w →K is the canonical symmetrizing form.

We denote by τ :OG^F → O the canonical symmetrizing form. We denote by τ_H the restriction of |U^F_G|τ toH^G: it is a symmetrizing form on H^G (recall that |U^F_G| is invertible in O and is the highest power of p dividing |G^F|). Note that

τ_H(e_ψ) = 1.

Of course, the extension KτH : KH^G → K is a symmetrizing form on KH^G. We have

(3.5) KτH = ˜τ ◦_KCur^G.

Proof. Sinceτw is a class function on T^F_w, we have, by Remark 2.6,

˜

τ(_KCur^G(h)) = 1

|W| X

w∈W

R^G_Tw(τw)(h)

for allh∈KH^G. But, by [DM, Proposition 12.9 and Corollary 12.14], we have 1

|W| X

w∈W

R_T^G_w(τw) =|U^F_G|τ.

This completes the proof of the formula 3.5.

3.C. On the image of Cur^G. We are not able to determine in general the sub-O- algebra Im(Cur^G) of Q

w∈W OT^F_w. Of course, we have (3.6) Im(Cur^G)⊂Im(_KCur^G)∩ Y

w∈W

OT^F_w . However, there are cases where this inclusion is an equality:

Theorem 3.7. If ℓ does not divide the order of W, then Im(Cur^G) = Im(_KCur^G)∩ Y

w∈W

OT^F_w .

Proof. Let A be the image of Cur^G. Then, since H^G is a symmetric algebra (with symmetrizing form τH), it follows from 3.5 that A is a symmetric algebra (with symmetrizing form ˜τA, the restriction of ˜τ toA).

Now, let B = Im(_KCur^G)∩ Q

w∈WOT^F_w

. If ℓ does not divide |W|, then the restriction of ˜τ toB defines a map ˜τ_B :B → O. By construction, we have A⊂B ⊂ KA. So the result follows from Lemma 3.8 below.

Lemma 3.8. Let (A, τ) be a symmetric O-algebra and let B be a subring of KA such that A ⊂ B and _Kτ(B)⊂ O. Then A =B.

Proof. Let (a₁, . . . , a_n) be an O-basis of A and let (a^∗₁, . . . , a^∗_n) denote the dual O- basis of A (with respect toτ). Then, for all h∈KA, we have h=Pn

i=1Kτ(ha^∗_i)a_i. Now, if moreover h∈ B, then ha^∗_i ∈ B for all i, so Kτ(ha^∗_i)∈ O. So h∈ A.

Remark 3.9 - Ifℓ does not divide the order ofW, then the Sylow ℓ-subgroups of G^F are abelian. If G^F =SL2(Fq), if q is odd and if ℓ= 2, then the inclusion 3.6 is strict. IfG^F =GL3(F2) and ifℓ = 3, thenℓdivides |W|but the Sylow 3-subgroups of G^F are abelian: in this case, a brute force computation shows that the inclusion 3.6 is an equality. This suggests the following question: do we have an equality in 3.6 if and only if the Sylow ℓ-subgroups of G^F are abelian?

By Corollary 3.4 and Theorem 3.7, we get:

Corollary 3.10. Let t= (t_w)∈Q

w∈W OT^F_w and assume that ℓ does not divide the order of W. Then t∈Im Cur^G if and only if, for all (w, θ), (w^′, θ^′)∈ S^G such that (w, θ)≡(w^′, θ^′), we have θ(tw) =θ^′(tw^′).

Corollary 3.11. Let h∈KH^G and assume that ℓ does not divide the order of W. Then h∈ H^G if and only if _KCur^G_T(h)∈ OT^F for all F-stable maximal tori of G.

The next result has been announced at the beginning of §2.C.

Corollary 3.12. If L is anF-stable Levi subgroup of a parabolic subgroup ofGand if ℓ does not divide the order of W, then _KCur^G_L(H^G)⊂ H^L.

Proof. Let h ∈ H^G and let h^′ = _KCur^G_L(h). By Corollary 3.11, it is sufficient to show that_KCur^L_T(h^′)∈ OT^F for allF-stable maximal torusTofL. But this follows from the transitivity of the Curtis maps (see Proposition 2.2) and from the fact that

KCur^G_T(H^G)⊂ OT^F (see Theorem 2.7).

3.D. Truncation at unipotent blocks. We keep the notation introduced in§2.D:

for instance, b^G denotes the sum of the unipotent blocks of G^F.

Theorem 3.13. Assume that ℓ does not divide the order of W. Let S denote a Sylow ℓ-subgroup of G^F and let T denote a maximally split F-stable maximal torus of CG(S). Then Cur^G_T induces an isomorphism

b^GH^G ≃(OT^Fb^T)^NGF(T) ≃(OS)^NGF(S).

Proof. First, sinceℓdoes not divide the order ofW,S is contained in some maximal torus and the centralizerCG(S) is anF-stable Levi subgroup of a parabolic subgroup of G. In particular, S is abelian, T^F contains S and S is a Sylow ℓ-subgroup of T^F. This implies that N_G^F(T) ⊂ N_G^F(S). Moreover, if n ∈ N_G^F(S), then ⁿT is another maximally split maximal torus of C_G(S) so there exists g ∈ C_G^F(S) such that ⁿT= ^gT. This shows that

(1) NG^F(S) =NG^F(T).CG^F(S).

This also implies that the map OS → OT^Fb^T,x7→xb^T induces an isomorphism (OT^Fb^T)^NGF(T) ≃(OS)^NGF(S).

So we only need to show that Cur^G_T induces an isomorphism of algebras b^GH^G ≃ (OT^Fb^T)^NGF(T).

Now, by 2.8, we have that Cur^G_T(b^GH^G) ⊂ (OT^Fb^T)^NGF(T). So it remains to prove that Cur^G_T is injective on b^GH^G and that the above inclusion is in fact an equality.

Let us first prove that Cur^G_T is injective on b^GH^G. Let T^∗ denote an F^∗-stable maximal torus which is dual to T. Let S^∗ denote the Sylow ℓ-subgroup of T^∗F∗. Then |G^F| = |G^∗F∗| and |T^F| = |T^∗F∗| so S^∗ is a Sylow ℓ-subgroup of G^∗F∗. In particular, every ℓ-element of G^∗F∗ is conjugate to an element of S^∗. So _KCur^G_T is injective on b^G₁ KH^G, as desired.

Moreover, since S^∗ is abelian, two elements of S^∗ are conjugate in G^∗F∗ if and only if they are conjugate underN_G^∗F∗(S^∗) that is, if and only if they are conjugate under N_G^∗F∗(T^∗): indeed, by the same argument used above for proving (1), we have

(1^∗) N_G^∗F∗(S^∗) =N_G^∗F∗(T^∗).CG^∗(S^∗).