2 Generalities on the Satake homomorphisms

Comparison of compact induction with parabolic induction

Henniart Guy, Vigneras Marie-France July 17, 2012

Abstract

Let F be a non-archimedean locally compact field of residual characteristic p, G a reductive connected F-group and K a special parahoric subgroup of G(F). We choose a parabolic F-subgroup P of G with Levi decomposition P = M N in good position with respect to K. Let C be an algebraically closed field of characteristic p, and V an irreducible smooth C-representation of K. We investigate the natural intertwiner from the compact induced representation c-Ind^G(F)_K V to the parabolic induced representa- tion Ind^G(F_P(F⁾₎(c-Ind^M(F)_M(F)∩KV_N(F)∩K). Under a regularity condition on V, we show that the intertwiner becomes an isomorphism after localisation at a specific Hecke operator.

When F has characteristic 0, G is F-split and K is hyperspecial, the result was essen- tially proved by Herzig. We define the notion of K-supersingularity for an irreducible smooth C-representation of G(F) which extends Herzig’s definition for admissible irre- ducible representations and we give a list of irreducible representations which are neither supercuspidal nor K-supersingular.

Contents

1 Introduction 1

2 Generalities on the Satake homomorphisms 4

3 Representations of G(k) 9

4 Representations of G(F) 13

4.1 Notations . . . . 13 4.2 S⁰ is a localisation . . . . 13 4.3 Decomposition of the intertwiner . . . . 15

5 Hecke operators 16

5.1 Definition of Hecke operators . . . . 17 5.2 Compatibilities between Hecke operators . . . . 17

6 Proof of the main theorem 21

7 Supersingular representations of G(F) 24

1 Introduction

LetFbe a non-archimedean locally compact field of residual characteristicp,Ga reductive connectedF-group andCan algebraically closed field of characteristicp. We are interested

in smooth admissibleC-representations ofG(F). Two induction techniques are available, compact induction c-Ind^G(F)_K from a compact open subgroup K of G(F) and parabolic induction Ind^G(F)_P(F) from a parabolic subgroup P(F) with Levi decomposition P(F) = M(F)N(F) . Here we want to investigate the interaction between the two inductions.

More specifically assume thatG(F) =P(F)KandP(F)∩K= (M(F)∩K)(N(F)∩K).

We construct (Def. 2.1) for any finite dimensional smooth C-representation V of K, a canonical intertwiner

IV : c-Ind^G(F)_K V →Ind^G(F)_P(F)(c-Ind^M(F)_M(F)∩KV_N(F)∩K),

where V_N(F)∩K stands for the N(F)∩K-coinvariants in V, and a canonical algebra homomorphism

S⁰ :H(G(F), K, V)→ H(M(F), M(F)∩K, V_N(F)∩K),

where as in [HV], the Hecke algebra H(G(F), K, V) is End_G(F)c-Ind^G(F_K ⁾V seen as an algebra of double cosets of Kin G, and similarly forH(M(F), M(F)∩K, V_N(F)∩K). By construction

(IV(Φ(f)))(g) =S⁰(Φ)(IV(f)(g)), forf ∈c-Ind^G(F)_K V,Φ∈ H(G(F), K, V), g∈G(F).

LetV^∗ be the contragredient representation of V. We constructed in [HV] a Satake homomorphism

S:H(G(F), K, V^∗)→ H(M(F), M(F)∩K,(V^∗)^N(F)∩K).

Here we show that S⁰ andS are related by a natural anti-isomorphism of Hecke algebras (Proposition 2.3).

We studyIV further in the particular case whereK a special parahoric subgroup and V is irreducible. Such aV is trivial on the pro-p-radicalK+ ofK. The quotientK/K+ is the group ofk-points of a connected reductivek-groupGk, so that we can use the theory of finite reductive groups in natural characteristic. We write K/K+ =G(k). The image of P(F)∩K =P0 in G(k) is the group of k-points of a parabolic subgroup of Gk. We write P0/P0∩K+ =P(k), and we use similar notations for M andN, for the opposite parabolic subgroup P =M N (Section 4.1), and for a minimal parabolic F-subgroup B ofGcontained inP, of Levi decompositionB=ZU.

We say that V is P-regular when the stabilizer PV(k) in G(k) of the line V^U(k) is contained in P(k) (this does not depend on the choice ofB). An equivalent definition is that, forh∈Kwhich does not belong toP0P0, the kernel of the quotient mapV →V_N(k) containshV^N(k)(Def. 3.7 and Cor. 3.20).

We choose a maximalF-split torusS in M such thatK stabilizes a special vertex in the apartment ofG(F) associated to S. We choose an elements∈S(F) which is central in M(F) and strictlyN-positive, in the sense that conjugation bysstrictly contracts the compact subgroups ofN(F). There is a unique Hecke operatorT_M inH(M(F), M₀, V_N(k)) with support inM₀sand value atsthe identity ofV_N(k). We prove (Prop.4.5):

Proposition 1.1. The mapS⁰ is a localisation atT_M.

This means thatS⁰ is injective, that T_M belongs to the image of S⁰, is central and invertible inH(M(F), M₀, V_N(k)), and that

H(M(F), M₀, V_N(k)) =S⁰(H(G(F), K, V))[T_M⁻¹].

This is a consequence of the analogous property ofS proved in [HV].

In this particular case, following a suggestion of Abe, we show thatI_V is injective. We introduce the localisation Θ ofI_V atT_M. AsI_V is injective, its localisation Θ is injective.

Our main theorem is:

Theorem 1.2. (Theorem 4.6) The map

Θ :H(M(F), M0, V_N(k))⊗_{H(G(F),K,V),S}0c-Ind^G(F)_K V →Ind^G(F_P(F⁾₎(c-Ind^M_M^(F)_(F)∩KV_N(k)) is bijective if and only if V isP-regular.

This result was proved by Herzig [Herzig], [Abe], when F has characteristic 0, G is F-split andK is hyperspecial (in this case the Hecke algebras are commutative).

Writing Z_G(V) for the center of H(G(F), K, V) and Z_M(V_N(k)) for the center of H(M(F), M(F)∩K, V_N(k)), the theorem implies by specialisation:

Corollary 1.3. IfV isP-regular, for any rightZ_M(V_N(k))-moduleχ, the representations of G(F)

χ⊗_ZG(V),S⁰c-Ind^G(F)_K V and Ind^G(F)_P(F)(χ⊗_ZM(VN(k))c-Ind^M(F)_M(F)∩KV_N(k)) are isomorphic.

To prove the theorem, we follow the method of Herzig and we decompose I_V as the compositeI_V =ζ◦ξof twoG(F)-equivariant maps, the natural inclusionξof c-Ind^G(F)_K V in c-Ind^G(F_K ⁾(c-Ind^G(k)_P(k)V_N(k)), and the natural map

ζ: c-Ind^G(F)_K (c-Ind^G(k)_P(k)VN(k))→Ind^G(F)_P(F)(c-Ind^M(F)_M(F)∩KVN(k)),

associated to the quotient map c-Ind^G(k)_P(k)V_N(k) → V_N(k) (see (2) below). We write P for the parahoric subgroup inverse image of P(k) in K and TP for the Hecke operator in H(G(F),P, VN(k)) with supportPsP and value at s the identity of VN(k). With no regularity assumption onV we prove

ζ◦T_P =T_M◦ζ .

Seeing c-Ind^G(F)_K (c-Ind^G(k)_P(k)V) = c-Ind^G(F)_P V_N(k)and Ind^G(F)_P(F)(c-Ind^M(F)_M(F)∩KV_N(k)) asC[T]- modules via T_P andTM, the mapζ isC[T]-linear and we prove (Corollary 6.5):

Theorem 1.4. The localisation atT ofζ is an isomorphism.

To studyξ, we consider the Hecke operatorTG inH(G(F), K, V) with support KsK and value at s the natural projector V → V^N(k), and the Hecke operator T_K,P from c-Ind^G(F)_P V_N(k) to c-Ind^G(F_K ⁾V with support KsP and value at s given by the natural isomorphism V_N(k)→V^N(k). With no regularity assumption on V we prove

T_K,P◦ξ=TG . Assuming thatV isP-regular we prove:

ξ◦T_K,P =T_P S⁰(T_G) =T_M .

Seeing c-Ind^G(F)_K V as aC[T]-module viaTG = (S⁰)⁻¹(TM), the mapξisC[T]-linear and:

Theorem 1.5. The localisation atT ofξis an isomorphism if and only ifV isP-regular.

Our main theorem 1.2 follows from these two theorems.

As in Herzig and Abe, we define the notion ofK-supersingularity.

Definition 1.6. We say that an irreducible smooth C-representation π of G(F) is K- supersingular when

H(M(F), M0, V_N(k))⊗_{H(G(F),K,V),S}⁰ Hom_G(F)(c-Ind^G(F)_K V, π) = 0

for any irreducible smooth C-representationV ofKand any standard Levi subgroupM 6=

G.

The properties ofK-supersingularity and of supercuspidality (not being a subquotient of Ind^G(F)_P(F)τ for some irreducible smoothC-representationτofM(F)6=G(F)) are equiva- lent whenGisF-split,Kis hyperspecial and the characteristic ofF is 0. We see our main theorem as the first step towards the classification of irreducible smoothC-representations ofG(F) in terms of supersingular ones. We obtain a partial result in this direction:

Theorem 1.7. Letπ be an irreducible smoothC-representation ofG(F).

i. Ifπ is isomorphic to a subrepresentation or is an admissible quotient ofInd^G(F)_P(F)τ as above, then πis not K-supersingular.

ii. Ifπis admissible and

(1) H(M(F), M0, V_N(k))⊗_{H(G(F),K,V),S}⁰ Hom_G(F)(c-Ind^G(F)_K V, π)6= 0

for some Q-coregular irreducible subrepresentationV of π|K and some standard parabolic subgroups P =M N ⊂Q=LN⁰6=G, thenπis not supercuspidal.

We would like to thank Noriyuki Abe for his helpful remarks which in particular saved us from a blunder in the example 5.6.

2 Generalities on the Satake homomorphisms

In this chapter we give a functorial construction of Herzig’s Satake transformS⁰in a rather general situation. Let C be a field, G a locally profinite group,K an open subgroup of G and P a closed subgroup of G satisfying “the Iwasawa decomposition” G = KP. We choose a smooth C[K]-module V. As in [HV], we assume that P is the semi-direct product of a closed invariant subgroupN and of a closed subgroupM, and thatK∩P is the semi-direct product of N∩K byM ∩K. We also impose :

(A1) Each double cosetKgK in Gis the union of a finite number of cosetsKg⁰ and the union of a finite number of cosetsg⁰⁰K(the first condition is equivalent to the second by taking inverses).

(A2) V is a finite dimensionalC-vector space.

The smoothC[K]-moduleV gives rise to a compactly induced representation c-Ind^G_KV and a smoothC[P]-moduleWgives rise to the full smooth induced representation Ind^G_PW. We consider the space of intertwiners

J := Hom_G(c-Ind^G_KV,Ind^G_PW).

By Frobenius reciprocity for compact induction (as K is open in G), the C-module J is canonically isomorphic to HomK(V,Res^G_KInd^G_PW); to an intertwiner I we associate the functionv 7→I[1, v]K where [1, v]K is the function in c-Ind^G_KV with supportK and valuev at 1. By the Iwasawa decomposition and the hypothesis thatKis open inG, we get by restricting functions to K an isomorphism of C[K]-modules from Res^G_KInd^G_PW onto Ind^K_P∩K(Res^P_P∩KW). Using now Frobenius reciprocity for the full smooth induction Ind^K_P∩K from P∩K toK, we finally get a canonicalC-linear isomorphism

J 'Hom_P∩K(V, W)

(we now omit mentioning the obvious restriction functors in the notation); this map associates to an intertwiner Ithe function v7→(I[1, v]_K)(1).

We could have proceeded differently, first applying Frobenius reciprocity to Ind^G_PW, gettingJ 'HomP(c-Ind^G_KV, W), then identifying Res^G_Pc-Ind^G_KV with c-Ind^P_P∩KV, and finally applying Frobenius reciprocity to c-Ind^P_P∩KV. In this way we also obtain an iso- morphism of J onto Hom_P∩K(V, W), which is readily checked to be the same as the preceding one.

Assume also that W is a smooth C[M]-module, seen as a smooth C[P]-module by inflation. Then Ind^G_PW is the ”parabolic induction” ofW, and HomP∩K(V, W) identifies with HomM∩K(VN∩K, W), whereVN∩K is the space of coinvariants ofN∩K inV. With that identification, an intertwinerIis sent to the map fromVN∩K toW sending the image v ofv∈V inV_N∩K to (I[1, v]_K)(1). By Frobenius reciprocity again Hom_M∩K(V_N∩K, W) is isomorphic to Hom_M(c-Ind^M_M∩KV_N∩K, W), so overall we obtain an isomorphism (2) j:J = Hom_G(c-Ind^G_KV,Ind^G_PW)→Hom_M(c-Ind^M_M∩KV_N∩K, W),

which associates to I ∈ J the C[M]-linear map sending [1, v]_M∩K to (I[1, v]_K)(1).

The reciprocal isomorphism sends I⁰ ∈ HomM(c-Ind^M_M∩KV_N∩K, W) to the element in HomG(c-Ind^G_KV,Ind^G_PW) which for v ∈ V, sends [1, v]K to the unique function with valueI⁰([1, kv]M∩K) atk∈K.

ForW = c-Ind^M_M∩KV_N∩K the isomorphismj is writtenjV,

jV : HomG(c-Ind^G_KV,Ind^G_P(c-Ind^M_M∩KVN∩K))→EndM(c-Ind^M_M∩KV_N∩K). Definition 2.1. We define IV in HomG(c-Ind^G_KV,Ind^G_P(c-Ind^M_M∩KVN∩K)) such that jV(IV)is the unit element ofEndM(c-Ind^M_M∩KVN∩K). The intertwiner IV is determined by the condition

(3) (I_V[1, v]_K)(1) = [1, v]_M∩K

for all v∈V.

The isomorphismj is natural inV andW. The functor FV :W 7→HomG(c-Ind^G_KV,Ind^G_PW)

from the category of smooth C[M]-modules to the category of sets is representable by c-Ind^M_M∩KVN∩K. Let now V⁰ be another finite dimensional smooth C[K]-module. Any G-intertwiner

b: c-Ind^G(F)_K V →c-Ind^G(F)_K V⁰

gives a morphism of functors FV⁰ → FV. By representatibility of FV and FV⁰ there is then a uniqueC[M]-morphism

S⁰(b) : c-Ind^M_M∩KV_N∩K→c-Ind^M_M∩KV_N⁰_∩K such that the following diagram

(4) HomG(c-Ind^G_KV⁰,Ind^G_PW) ^j

0 //

I⁰7→I⁰◦b

HomM(c-Ind^M_M∩KV_N∩K⁰ , W)

I⁰7→I⁰◦S⁰(b)

HomG(c-Ind^G_KV,Ind^G_PW)

j //HomM(c-Ind^M_M∩KVN∩K, W)

is commutative for all smooth C[M]-modulesW (the horizontal maps are the canonical isomorphisms constructed above, the vertical maps are given by composition with b or withS⁰(b)). TakingW = c-Ind^M_M∩KV_N∩K⁰ we get

(5) S⁰(b) =jV(IV⁰◦b).

IfV⁰ is a third finite dimensional smoothC[K]-module and b⁰: c-Ind^G(F)_K V⁰ →c-Ind^G(F)_K V⁰⁰

is a G-intertwiner, then b⁰◦b : c-Ind^G(F)_K V → c-Ind^G(F_K ⁾V⁰⁰ is a G-intertwiner and we have obviously

(6) S⁰(b⁰◦b) =S⁰(b⁰)◦ S⁰(b). TakingV =V⁰ =V⁰⁰ we get an algebra homomorphism

S⁰ : EndG(c-Ind^G_KV)→EndM(c-Ind^M_M∩KVN∩K) such that

j(I◦b) =j(I)◦ S⁰(b) forI in HomG(c-Ind^G_KV,Ind^G_PW).

By the naturality ofj in W, for any homomorphismα:W⁰ →W of smooth C[M]- modules we have a commutative diagram

Hom_G(c-Ind^G_KV,Ind^G_PW⁰) ^j //

Ind(α)

Hom_M(c-Ind^M_M∩KV_N∩K, W⁰)

α

HomG(c-Ind^G_KV,Ind^G_PW)

j //HomM(c-Ind^M_M∩KVN∩K, W)

for any V. ForW =W⁰ we obtainj((Ind^G_Pa)◦I) =a◦j(I) fora∈EndM(W). We have j_V((Ind^G_Pα)◦I_V) =α

for allαin HomM(c-Ind^M_M∩KVN∩K, W). WhenW =W⁰= c-Ind^M_M∩KVN∩K we deduce I_V ◦b= (Ind^G_PS⁰(b))◦I_V

forb∈EndG(c-Ind^G_KV), by applyingj_V⁻¹ to (5).

We now want to interpret the previous results in terms of actions of Hecke algebras. By Frobenius reciprocity HomG(c-Ind^G_KV,c-Ind^G_KV⁰) identifies with HomK(V,Res^G_Kc-Ind^G_KV⁰), as aC-module; to aG-intertwinerbwe associate the mapv7→bv :=b([1, v]K). From such a b, we get a map

Φ_b:G→Hom_C(V, V⁰) , g7→(v7→b_v(g)).

In this way we identify HomG(c-Ind^G_KV,c-Ind^G_KV⁰) with the spaceH(G, K, V, V⁰) of func- tions Φ fromGto HomC(V, V⁰) such that:

(i) Φ(k⁰gk) =k⁰◦Φ(g)◦kfork, k⁰ inK,g inG, where we have writtenk, k⁰ for the endomorphisms v7→kv, v⁰ 7→k⁰v⁰ ofV and ofV⁰;

(ii) The support of Φ is a finite union of double cosetsKgK.

The natural map H(G, K, V, V⁰)×c-Ind^G_KV →c-Ind^G_KV⁰ is given by convolution Φ∗f(g) = X

h∈G/K

Φ(h)(f(h⁻¹g)) = X

h∈K\G

Φ(gh⁻¹)(f(h)).

The composition

H(G, K, V⁰, V⁰⁰)× H(G, K, V, V⁰)→ H(G, K, V, V⁰⁰) corresponding to the composition of intertwiners is given by convolution

Φ∗Ψ(g) = X

h∈G/K

Φ(h)Ψ(h⁻¹g) = X

h∈K\G

Φ(gh⁻¹)Ψ(h)

(the term Φ(h)Ψ(h⁻¹g)(v) vanishes, for fixedg ∈ G and v ∈ V, outside finitely many cosets Kh, so that the sum makes sense). The map

b7→ S⁰(b) : HomG(c-Ind^G_KV,c-Ind^G_KV⁰)→HomM(c-Ind^M_M∩KV_N∩K,c-Ind^M_M∩KV_N⁰_∩K) translates into a map

S⁰ : H(G, K, V, V⁰)→ H(M, M ∩K, VN∩K, V_N∩K⁰ ).

The following proposition shows that our definition of S⁰ is equivalent to Herzig’s defini- tion.

Proposition 2.2. The homomorphismS⁰:H(G, K, V, V⁰)→ H(M, M∩K, VN∩K, V_N⁰_∩K) is given by

S⁰(Φ)(m)(v) = X

n∈(N∩K)\N

Φ(nm)(v) for m∈M, v∈V , where bars indicate the image in V_N∩K of elements in V and similarly for V⁰.

Proof. Letb ∈Hom_G(c-Ind^G_KV,c-Ind^G_KV⁰) and Φ_b ∈ H(G, K, V, V⁰) corresponding to b.

We have (5)

S⁰(Φ_b) = Φ_S0(b)= Φ_jV(I

V0◦b).

Forg∈G, v∈V, m∈M, we have Φb(g)(v) =b([1, v]K)(g) inV⁰ and

S⁰(Φb)(m)(v) = (jV(IV⁰◦b))([1, v]_M∩K)(m) = ((IV⁰◦b)([1, v]K)(1))(m) in V_N∩K⁰ . Using the Iwasawa decomposition we write in c-Ind^G_KV

b([1, v]K) =X

h

h⁻¹[1,Φb(h)(v)]K

forhrunning over a system of representatives of (P∩K)\P.

We compute now the elementI_V⁰(h⁻¹[1,Φ_b(h)(v)]_K)(1) of c-Ind^M_M∩KV_N⁰_∩K. AsI_V⁰ is G-equivariant we have in Ind^G_P(c-Ind^M_M∩KV_N⁰_∩K),

IV⁰(h⁻¹[1,Φb(h)(v)]K) =h⁻¹IV⁰([1,Φb(h)(v)]K) Taking the value at the unit element 1 of Gwe obtain

(h⁻¹IV⁰([1,Φb(h)(v)]K))(1) =IV⁰([1,Φb(h)(v)]K)(h⁻¹) =h⁻¹(IV⁰([1,Φb(h)(v)]K)(1)). Recalling (3), this is equal to

h⁻¹[1,Φb(h)(v)]_M∩K =m_h⁻¹[1,Φb(h)(v)]M∩K=m⁻¹_h [1,Φb(h)(v)]M∩K , where mhis the image ofhinM. We deduce

(IV⁰◦b)([1, v]K))(1) =X

h

m⁻¹_h [1,Φb(h)(v)]_M∩K

Formin a system of representatives of (M∩K)\M, andnin a system of representatives of (N∩K)\N, the elementsnmform a system of representatives of (P∩K)\P. We obtain

(I_V⁰◦b)([1, v]_K))(1) = X

m∈(M∩K)\M

m⁻¹[1, w_m]_M∩K , w_m:= X

n∈(N∩K)\N

Φ_b(nm)(v).

In [HV] we constructed a Satake homomorphism

S :H(G, K, V, V⁰)→ H(M, M∩K, V^N∩K, V^0N∩K) , S(Φ)(m)(v) = X

n∈N/(N∩K)

Φ(mn)(v),

for v∈V^N∩K. To compare S⁰ withS we need to take the dual. Remark thatK acts on the dual space V^∗ = Hom_C(V, C) ofV via the contragredient representation, and that the dual ofV^∗ is isomorphic toV by our finiteness hypothesis onV. It is straightforward to verify that the map

ι:H(G, K, V^0∗, V^∗)→ H(G, K, V, V⁰) , ι(Φ)(g) := (Φ(g⁻¹))^t, where the upper indext indicates the transpose, is aC-isomorphism and satisfies

ι(Φ∗Ψ) =ι(Ψ)∗ι(Φ) for Φ∈ H(G, K, V^0∗, V^∗),Ψ∈ H(G, K, V^00∗, V^0∗).

The linear forms onV which are (N∩K)-fixed identify with the linear forms onVN∩K, (V_N∩K)^∗'(V^∗)^N∩K ,

and similarly forV⁰ andV⁰⁰. This leads to a naturalC-linear isomorphism ιM :H(M, M∩K,(V^0∗)^N∩K,(V^∗)^N∩K)→ H(M, M∩K, V_N∩K, V_N⁰_∩K).

The following proposition describes the relation between the Satake homomorphisms S attached to V^0∗, V^∗ andS⁰ attached toV, V⁰.

Proposition 2.3. The following diagram is commutative H(G, K, V^0∗, V^∗) ^S//

ι

H(M, M ∩K,(V^0∗)^N∩K),(V^∗)^N∩K)

ι_M

H(G, K, V, V⁰) ^S

0 //H(M, M∩K, VN∩K, V_N⁰_∩K).

Proof. For Φ∈ H(G, K, V^0∗, V^∗), m∈M andv∈V of image vin VN∩K we have:

((ιM ◦ S)Φ)(m)(v) = (S(Φ)(m⁻¹))^t(v) = X

n∈N/(N∩K)

(Φ(m⁻¹n))^t(v)

= X

n∈(N∩K)\N

(Φ((nm)⁻¹))^t(v) = X

n∈(N∩K)\N

ι(Φ)(nm)(v) = ((S⁰◦ι)Φ)(m)(v).

By this proposition, the Satake mapS is injective if and only if the mapS⁰is injective because the maps ιandι_M are isomorphisms.

Proposition 2.4. Let V be a finite dimensional smooth C-representation of K. If the homomorphisms S⁰ :H(G, K, V⁰, V)→ H(M, M ∩K, V_N⁰_∩K, V_N∩K) are injective for all irreducible C-smooth representationsV⁰ ofK, then the interwiner

IV : c-Ind^G_KV →Ind^G_P(c-Ind^M_M∩KV_N∩K) is injective.

Proof. Assume that I_V is not injective. Then the kernel of I_V is a non-zero subrepre- sentation of c-Ind^G_KV, and contains an irreducible smooth C[K]-representation V⁰. By Frobenius reciprocity we get a non-zero intertwinerb∈Hom_G(c-Ind^G_KV⁰,c-Ind^G_KV) such thatI_V◦b= 0. By assumption, the mapS⁰ :H(G, K, V⁰, V)→ H(M, M∩K, V_N∩K⁰ , V_N∩K) is injective By the relation (5), this means that the map

b7→IV ◦b: HomG(c-Ind^G_KV⁰,c-Ind^G_KV)→HomM(c-Ind^M_M∩KV_N⁰_∩K,c-Ind^M_M∩KVN∩K). is injective, which gives a contradiction.

This criterion for the injectivity ofI_V was communicated to us by Noriyuki Abe.

3 Representations of G(k)

LetCbe an algebraically closed field of positive characteristicp, letkbe a finite field of the same characteristicpand of cardinalityq, and letGbe a connected reductive group over k. We fix a minimal parabolick-subgroupB ofGwith unipotent radicalU and maximal k-subtorusT. LetSbe the maximalk-split subtorus ofT, letW =W_G=W(S, G) be the Weyl group, let Φ = ΦG be the roots ofS with respect toU (called positive), ∆⊂Φ the subset of simple roots. For a∈Φ, let Ua be the unipotent subgroup denoted in ([BTII]

5.1) by U_(a). A parabolick-subgroupP ofGcontaining B is called standard, and has a unique Levi decompositionP =M N with Levi subgroupM (called standard) containing T. The standard parabolic subgroupP =M U =U M is determined byM. There exists a unique subset ∆M ⊂∆ such thatM is generated byT, Ua, U_−a forain the subset ΦM

of Φ generated by ∆M. This determines a bijection between the subsets of ∆ and the standard parabolick-subgroups ofG.

LetB =T U be the opposite ofB =T U, andP =M N the opposite ofP. We have B =w0Bw₀⁻¹ wherew0=w⁻¹₀ is the longest element of W. The roots of S with respect to U, i.e. the positive roots for U, are the negative roots for U. The simple roots forU are the roots−afora∈∆.

Fora∈∆ letG_a ⊂Gbe the subgroup generated by the unipotent subgroupsU_a and U_−a, and letT_a:=G_a∩T.

Definition 3.1. Let ψ:T(k)→C^∗ be a C-character of T(k). We denote by

∆ψ := {a∈∆ |ψ(Ta(k)) = 1}

the set of simple roots asuch that ψ is trivial onTa(k).

Example 3.2. G=GL(n) andS is the diagonal group. ThenT =S and the groupsTa

for a∈ ∆ are the subgroups Ti ⊂T for 1≤i ≤n−1, with coefficients xi =x⁻¹_i+1 and xj= 1 otherwise. Whenk=F2 is the field with 2 elements,T(k) is the trivial group.

Let V be an irreducible C-representation of G(k). When P = M N is a standard parabolic subgroup ofG, we recall that the natural action ofM(k) onV^N(k)is irreducible ([CE] Theorem 6.12). In particular, taking the Borel subgroupB=T U, the dimension of the vector spaceV^U(k)is 1 and the groupT(k) acts on V^U(k) by a characterψ_V.