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THE PRO-p-IWAHORI HECKE ALGEBRA OF A REDUCTIVE p-ADIC GROUP III (SPHERICAL HECKE ALGEBRAS AND SUPERSINGULAR MODULES)

Marie-France Vigneras

Journal of the Institute of Mathematics of Jussieu / FirstView Article / August 2015, pp 1 - 38 DOI: 10.1017/S1474748015000146, Published online: 03 June 2015

Link to this article: http://journals.cambridge.org/abstract_S1474748015000146 How to cite this article:

Marie-France Vigneras THE PRO-p-IWAHORI HECKE ALGEBRA OF A REDUCTIVE p-ADIC GROUP III (SPHERICAL HECKE ALGEBRAS AND SUPERSINGULAR MODULES). Journal of the Institute of Mathematics of Jussieu, Available on CJO 2015 doi:10.1017/S1474748015000146 Request Permissions : Click here

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THE PRO-p-IWAHORI HECKE ALGEBRA OF A REDUCTIVE p-ADIC GROUP III (SPHERICAL HECKE ALGEBRAS AND

SUPERSINGULAR MODULES)

MARIE-FRANCE VIGNERAS

UMR 7586, Institut de Mathematiques de Jussieu, 4 place Jussieu, Paris 75005, France ([email protected])

(Received26May2014; accepted30March2015)

Abstract LetRbe a large field of characteristicp. We classify the supersingular simple modules of the pro-p-Iwahori Hecke R-algebra H of a general reductive p-adic group G. We show that the functor of pro-p-Iwahori invariants behaves well when restricted to the representations compactly induced from an irreducible smooth R-representation ⇢ of a special parahoric subgroup K of G. We give an almost-isomorphism between the center ofHand the center of the spherical Hecke algebraH(G,K,⇢), and a Satake-type isomorphism forH(G,K,⇢). This generalizes results obtained by Ollivier for Gsplit and Khyperspecial toGgeneral andK special.

Keywords: group theory and generalizations; number theory 2010Mathematics subject classification: Primary 20C08

Secondary 22E50

Contents

1 Introduction 2

2 The characters ofhandH^aff 10

3 Distinguished representatives ofW₀\W 13

4 h-eigenspace in ⌘⌦hH 16

5 Centers 22

6 SupersingularH-modules 24

7 Pro-p-Iwahori invariants and compact induction 31

References 37

1. Introduction

Let pbe a prime number, let F be a finite extension ofQporFp((T)), and letG be the group of rational points of a connected reductive F-group.

1.1.

The smooth representations ofG over an algebraically closed fieldC of characteristic p have been the subject of many investigations in recent years, in the modulo pLanglands program. The pro-p-Iwahori invariant functor V 7!V^I(1) relates the representations of G to the modules of the pro-p-Iwahori Hecke C-algebra H=HC(G,I(1)) studied in [13–15]. The I(1)-invariant functor and the theory of H-modules play an increasingly important role in the representation theory of G modulo p. They are the key to the proof of the change of weight in the recent classification of irreducible smooth C-representations of G in terms of supersingular ones (a forthcoming work by Abe et al. [1]). The supersingular smooth irreducible C-representations ⇡ of G and their I(1)-invariant remain mysterious, but the supersingular simpleH-modules are classified in this paper, and the supersingularity of ⇡^I(1) and of ⇡ are related. A variant of the modulo p Langlands program seems to exist for H-modules. Grosse-Kloenne [5] constructed a functor from finite-dimensional HC(G L(n,Qp),I(1))-modules to finite-dimensional smooth C-representations of Gal_Qp, inducing a bijection between the simple supersingular HC(G L(n,F),I(1))-modules of dimension n and the irreducible smoothC-representations ofGal_F (the absolute Galois group ofF) of dimensionn as in [9,14].

In this paper, we prove that the I(1)-invariant functor behaves well when restricted to compactly induced representations c-Ind^G_K⇢, where ⇢ is an irreducible smooth C-representation of a special parahoric subgroup K of G. The vector space ⇢^I(1) has dimension 1, and the pro-p-Iwahori Hecke C-algebra h=HC(K,I(1)) of K acts on

⇢^I(1) by a character ⌘. The H-module (c-Ind^G_K⇢)^I(1) is isomorphic to ⌘⌦hH, and the spherical algebraEndCG(c-Ind^G_K⇢)is isomorphic to the algebraEnd_H(⌘⌦hH). This paper is devoted to the study of the modules ⌘⌦hH and of the spherical Hecke algebras End_H(⌘⌦hH). In the last section, we transfer our results fromHto the group G using the I(1)-invariant functor.

Let⇢ be an irreducible smooth C-representation of K, and let⌘,⌘₁be two arbitrary characters ofh. We obtain the following:

(i) Isomorphisms

(c-Ind^G_K⇢)^I(1)'⇢^I(1)⌦hH, EndCG(c-Ind^G_K⇢)'End_H(⇢^I(1)⌦hH).

(ii) A Satake-type isomorphism for the spherical Hecke algebraH(h,⌘)=End_H(⌘⌦h

H).

(iii) A basis of the space of intertwinersHom_H(⌘₁⌦hH,⌘⌦hH).

(iv) An almost-isomorphism from the center of H to the center of H(h,⌘) (an isomorphism between finite index affine subalgebras).

(v) The classification of the supersingular simpleH-modules.

WhenG is split and K hyperspecial, Ollivier proved (i), (ii), (iv) and (v). We follow her method. The alcove walk bases of H and the product formula [12, 15] allow us to simplify her method and to extend it to G general and K special. Analogs of 2, 3 were proved for G in [6,7] and 5 forG remains a wide-open question.

In the rest of this introduction, we consider the content of 2, 3, 4, 5.

After [13,14], a generalization of HC(G,I(1))was introduced in [12] when G is split, and in [15] for G general, in order to study it. This is an algebra HR(q_s,c_s˜) over a commutative ringR with two sets of parameters(qs), (c_s˜). The properties of this algebra are often proved by reduction to (q_s)=(1) (this changes the parameters (c_s˜)), and transferred toHR(0,c_s˜)by specialization to(q_s)=(0). The algebra HR(q_s,c_s˜)contains a natural finite-dimensional subalgebrah_R(qs,c_s˜).

In 1.2 and 1.3, we recall the basic properties of HR(q_s,c_s˜) used in this work and the dictionary between h_R(q_s,c_˜s),HR(q_s,c_s˜) and HR(K,I(1)),HR(G,I(1)) [15, 16].

Theorems 1.2,1.3, 1.4, and1.5are proved for h_R(0,c_s˜),HR(0,c_˜s), and are given in1.4.

They apply to the algebrasHR(K,I(1)),HR(G,I(1)) when R has characteristic p.

1.2.

LetW=(6,1,,3,⌫,W,Zk,W(1))be data consisting of the following:

(i) a reduced root system6 of basis1associated with the finite Weyl Coxeter system (W0,S)of an affine Weyl Coxeter system(W^aff,S^aff)acting on a real vector space V of dual of basis1, with aW₀-invariant scalar product;

(ii) three commutative groups,and3finitely generated, and Zk finite;

(iii) a groupW =W^affo=3oW0which is a semi-direct product of subgroups in two di↵erent ways,acting on(W^aff,S^aff)and W₀on3. The length`and the Bruhat order6of(W^aff,S^aff)extend trivially toW =W^affo;

(iv) a W₀-equivariant homomorphism ⌫:3!V such that the action of W^aff on V and the action of3on V by translationv7!v+⌫( )for 23, v2V, extend to an action of W by affine automorphisms permuting the set of affine hyperplanes H= {Ker(↵+n),|↵+n26^aff =6+Z};

(v) a system of the representatives ofW₀ in3:

3⁺:= {µ23|⌫(µ)2D⁺},

whereD⁺= {x2V |06↵(x), ↵21}is the dominant closed Weyl chamber;

(vi) an extension1! Z_k !W(1)!W !1.

Notation. The inverse image in W(1) of a subset X of W is denoted by X(1), and w˜ denotes an element of W(1) of image w2W. For c2 R[Zk], the conjugate of c by w˜ depends only onw, and is denotedw•c:= ˜wcw˜ ¹. The dominant Weyl chamberD⁺= {x2V |0<↵(x),↵21}is open. The dominant alcove C⁺ is the connected component D⁺\(V S

H2HH)of vertex02V. The set6^aff,+ of positive affine roots is the set of 26^aff positive onC⁺. The action ofW on V defines by functoriality an action of W on6^aff.

We will often suppose that3contains a subgroup3_T satisfying the following.

(T1) 3=F

y2Y3_Tyfor a finite setY. (T2) 3_T isW₀-stable.

(T3) There exists a central subgroup 3˜_T of 3(1) normalized by W₀(1) such that the quotient map3(1)!3 induces a group isomorphism3˜_T !^' 3_T sendingw˜µ˜w˜ ¹ towµw ¹ifw˜ 2W0(1)liftsw2W0andµ˜ 23˜_T liftsµ23_T.

Let(q_s˜,c_s˜)_s2S˜ aff(1))be a set of elements in R⇥R[Zk] satisfyingq_s˜0 =q_s˜,c_s˜0 =w•c_s˜ if

˜

s⁰= ˜ws˜w˜ ¹2S^aff(1),w˜ 2W(1), andq_ts˜=q_s˜,c_ts˜=tc_s˜ ift2 Zk. Asq_s˜ depends only on the images2S^aff ofs, we denote also˜ q_s˜=q_s.

There is a uniqueR-algebraH=HR(W,qs,c_s˜), free of basis(Tw_˜)_w˜2W(1), with product satisfying

(i) the braid relations:

Tw_˜Tw_˜0 =Tw_˜w˜0, ifw,˜ w˜⁰2W(1),`(w)+`(w⁰)=`(ww⁰), (1) allowing one to identifyR[(1)]to a subalgebra ofH;

(ii) the quadratic relations:

T_s˜T_˜s^⇤=q_ss˜², ifs˜2 S^aff(1),T_s˜^⇤=T_˜s c_s˜. (2) This is called the Iwahori–Matsumoto presentation of HR(W,qs,c_s˜).

The R-submodule of basis(Tw_˜)_w˜2W0(1) is a finite subalgebrah=h_R(W,qs,c_s˜).

The R-submodule of basis(Tw_˜)_w˜2Waff(1) is a subalgebraH^aff. The R-algebra H^aff is an algebra likeHwithtrivial, andHis isomorphic to the twisted tensor product

x⌦y7!xy:H^aff⌦^t_R[Zk]R[(1)]!H (3) of its subalgebrasR[(1)]andH^aff. The algebraHadmits an involutiveR-automorphism

◆, equal to the identity onR[(1)]and such that [15, Proposition 4.23]

◆(T_s˜):= T_s˜^⇤ fors2S^aff. (4) All the orientations that we consider are spherical [15]. For the orientationoassociated to an (open) Weyl chamberD_o, theo-positive side of the affine hyperplaneKer(↵+n)is the set of x2V where↵(x)+n>0, if↵26 takes positive values onD_o. The dominant orientationo, denoted byo⁺, is associated to the dominant Weyl chamberD⁺, and the anti-dominant orientation, denoted by o , to the anti-dominant Weyl chamber D⁺= D . The orientation associated to the Weyl chamber w ¹(D_o), w2W₀, is denoted by o•w. Forw2W of projectionw₀2W₀, the orientationo•w₀is also denoted by o•w.

We haveo• =ofor 23. We set

S_o^aff := {s2S^aff|C⁺ is in theo-positive side of Hs}, So:=S\S_o^aff, (5) where H_s is the affine hyperplane of V fixed by s and C⁺ the dominant alcove (Notation). There exists a unique set of bases (E_o(w))˜ _w˜2W(1) of H, parameterized by

the orientationso, satisfying [15,§5.3]

E_o(s)˜ :=T_s˜ ifs2S^aff S_o^aff, E_o(s)˜ :=T_s˜^⇤ifs2S_o^aff, (6) and the product formula, forw,˜ w˜⁰2W(1),

E_o(w)E˜ _o•w(w˜⁰)=E_o(w˜w˜⁰) if`(w)+`(w⁰)=`(ww⁰). (7) In particular, for ˜,˜⁰23(1),

E_o(˜)E_o(˜⁰)=E_o(˜ ˜⁰) if⌫( ),⌫( ⁰)belong to a same closed Weyl chamber. (8) We have Eo( )=T when⌫( )2Do.

The basis(Eo(w))˜ _w˜2W(1)is called an alcove walk basis; the alcove walk bases generalize the integral Bernstein bases defined in [11,14].

The R-submodule of basis (E_o(˜))_˜23(1) is a subalgebra Ao of H containing the subalgebraA⁺_o of basis(E_o(˜))_˜23+(1), isomorphic to R[3⁺(1)].

Ifqs =0for alls2S^aff, then forw,˜ w˜⁰2W(1)such that`(w)+`(w⁰) >`(ww⁰)we have Eo(w)˜ E_o•w(w˜⁰)=0; in particular,Eo(˜)Eo(˜⁰)=0if ˜,˜⁰23(1), and⌫( ),⌫( ⁰)do not belong to the same closed Weyl chamber.

1.3.

LetF be a local field of finite residue fieldkwithq elements and of characteristic p, and p_F a generator of the maximal ideal of the ring of integersO_F ofF. LetG,T,Z, andNbe respectively the F-rational points of a connected reductive F-group, a maximal F-split subtorus, its centralizer, and its normalizer. LetC⁺be an open alcove of the semi-simple apartment of Gdefined by T, let x₀be a special vertex of the closed alcove C⁺, and let I,I(1),K, be respectively the Iwahori subgroup ofGfixingC⁺, its pro-p-Sylow subgroup, and the parahoric subgroup of Gfixing x0.

We associate toG,T,Z,N,I,I(1),K the data

(W=(6,1,,3,⌫,W,Zk,W(1));(qs,c_s˜)),

and a group3_T, satisfying the properties given in§1.2with R=Z, as follows.

The apartment defined by T identifies with a Euclidean real vector space V. The set S^aff of orthogonal reflections with respect to the walls of C⁺generates an affine Coxeter system(W^aff,S^aff), given by a based reduced root system(6,1). The action ofN on the apartment transfers to an action onV. The subgroup Z acts by translations(z,x)7!x+

⌫_Z(z), (z,x)2Z⇥V, for an homomorphism⌫_Z :Z !V satisfying↵ ⌫_Z(t)= ↵(t)for t 2T and↵in the root system8ofT inG. There is a surjective map↵7!e↵↵:8!6, wheree↵ is a positive integer for all↵28.

Let T0:=T\K (the maximal compact subgroup of T), Z0:=K\Z (the parahoric subgroup of Z), and let Z₀(1)be the pro-p-Sylow subgroup of Z₀. Then

3_T :=T/T0, 3:=Z/Z0, 3(1):=Z/Z0(1), Zk:= Z0/Z0(1), W0:=N/Z, W :=N/Z0, W(1):=N/Z0(1).

The homomorphism ⌫_Z and the action of N on V are trivial on Z₀. They induce an homomorphism ⌫:3!V and an action of W on N. The monoid 3⁺ represents the

orbits ofW0in 3[7, 6.3] and the double cosets K\G/K. The groups W,W(1)represent the double cosets I\G/I,I(1)\G/I(1). The groupis the W-stabilizer of the alcoveC⁺. We denote byw˜ an element of W(1)of imagewin W, and we callw˜ a lift of w.

Fors2S^aff, letKs be the parahoric subgroup ofGfixing the face ofC⁺fixed bys. The quotient of Ks by its pro-p-radical is the group Gs,k of rational points of ak-reductive connected group of rank1. The image of I(1)inG_s,k is the groupU_s,k of rational points of the unipotent radical of ak-Borel subgroup Z_kU_s,kof opposite group Z_kU_s,k. It is known that s admits a lift ns 2N\Ks of image in Gs,k belonging to the group hUs,k,Us,ki generated byU_s,k[U_s,k. The image ofn_s inW(1)is called an admissible lift ofs. We set Z_k,s =Z_k\ hUs,k,U_s,ki.

Fors2 S^aff,s˜ an admissible lift ofs, andt 2 Zk, let

q_s = [I n_sI :I]is a power ofq, c_s :=(q_s 1)|Z_k,s| ¹ X

z2Zk,s

z, andc_ts˜=P

z2Zk,sc_s˜(z)tz, for positive integersc_s˜(z)=c_s˜(z ¹)of sumqs 1, constant on each coset modulo{xs(x) ¹|x2 Zk}, andc_s˜⌘cs mod p as in [15, Theorem 2.2].

The cocharacter group X_⇤(T) of T is isomorphic to 3_T and embeds in 3(1) by the map µ7!µ(p_F) ¹: X_⇤(T)!Z followed by the quotient maps of Z onto3and 3(1).

Remembering the sign in the definition of⌫,

µ23⁺_T ,↵(µ(p_F))2O_F for all↵21.

We identifyµ with its image in3_T, andµ˜ denotes its image in3(1).

For a commutative ring R, the pro-p-Iwahori Hecke R-algebra HR(G,I(1)) is isomorphic to the algebraHR(q_s,c_s˜)associated to this data.

The pro-p-Iwahori Hecke R-algebra HR(K,I(1))of K is a subalgebra of HR(G,I(1)) isomorphic to the finite subalgebrah(qs,c_s˜)ofH.

The Iwahori Hecke R-algebra HR(G,I) is an algebra Hassociated to the same data except that Zk = {1},W(1)=W,cs =qs 1.

The group G is split ,T =Z )3_T =3. The group G is quasi-split ,Z is the F-points of an F-torus)3(1)is commutative. The group G is semi-simple,Ker⌫ is finite)is finite and⌫ is injective on3_T.

The quotient of K by its pro-p-radical K(1)is the group Gk of k-rational points of a connected reductive k-group. The images in G_k of T₀, Z₀, I, and I(1) are the groups T_k,Z_k,B_k, and U_k of k-rational points of a maximal k-split torus, its centralizer (a k-torus), a Borel k-subgroup containing the maximal k-split torus, and its unipotent radical.

The finite Hecke algebrasHR(K,I(1))andHR(G_k,U_k)are isomorphic.

The conditionqs =0 for alls2 S^aff means that the characteristic of R is p. Then, c_ts˜= |Zk,s| ¹ X

z2Zk,s

tz,

and the irreducible smooth R-representations⇢ of K are trivial on K(1); they identify with the irreducible R-representations of G_k, in bijection with the characters of HR(G_k,U_k)by theU_k-invariant functor⇢ 7!⇢^Uk for R as in1.4.

1.4.

For the remainder of this article, unless otherwise specified, we are in the setting of§1.2 withqs =0for alls2S^aff, andRis a field containing a root of unity of order the exponent of Z_k.

Notation. We denote by Zˆk the group of R-characters of Zk. For a character 2 Zˆk, a character⌘of h, and a character4 ofH^aff, we set

S^aff:= {s2S^aff| (c_s˜)6=0}, S :=S^aff\S, (9) S⌘:= {s2 S|⌘(T_s˜)6=0}, S^aff₄ := {s2S^aff |4(T_s˜)6=0}. (10) These sets are independent of the choice of the lift s˜ of s. For (w, )˜ 2W(1)⇥Zˆk we denote by ^w 2Zˆk the character ^w(t)= (w˜tw˜ ¹)fort 2 Zk. The subgroup generated by a subset X of a group is denoted byhXi. For 23we set

1 := {↵21|↵ ⌫( )=0}, S := {s↵ |↵21 }. (11) We recall from §1.2 the R-algebra h associated to the finite Coxeter system (W₀,S) and the extension1!Z_k!W₀(1)!W₀!1, of basis(Tw_˜)_w˜2W0(1) satisfying the braid relations and the quadratic relations T_s˜(T_s˜ c_s˜)=0 fors˜2S(1).

Theorem 1.1(The characters ofh). (a) The characters⌘of hare in bijection with the pairs( ,J), where 2Zˆk and J⇢S , =⌘|Zk, and J=S⌘.

(b) For any ⌘, there exists an orientation o such that the equivalent properties S⌘= S \So,⌘(Eo(s˜))=0, for alls2S, hold true. We set o:=⌘.

(c) For two characters⌘₁,⌘ofh, there exists an orientationosuch that⌘₁=( ₁)_o,⌘=

o if and only if

S⌘\S ₁ =S⌘₁\S .

For a reduced decomposition of w˜ = ˜s₁. . .s˜_r of W(1), the element cw_˜ =c_s˜1. . .c_s˜r of R[Zk] does not depend on the choice of the reduced decomposition [15, Propositions 4.13(ii) and 4.22].

Theorem 1.2(A basis of the intertwiners). Let⌘₁,⌘be two characters ofhof restrictions

1, to Zk.

(a) ⌘₁ is contained in⌘⌦hH(is a submodule) if and only if

1= , S⌘₁\S =S⌘\S , for some 23⁺. (b) For 23⁺ satisfying (a), there exists a non-zero H-intertwiner

8_˜ :1⌦17!1⌦E˜ :⌘₁⌦hH!⌘⌦hH, E˜ := X

w₀2Y

1(cw_˜0) ¹⌦T_{˜ ˜w0}, whereY = {w₀2 hS 1 S⌘₁i| ₁^w0 = 1,`( w<sub>0</sub>)=`( ) `(w₀)}, andw˜₀is a lift of w₀; note that ₁(cw_˜0) ¹⌦T_{˜ ˜w0} does not depend on the choice of the lift.

(8_˜), for 23⁺ satisfying (a), is a basis of Hom_H(⌘₁⌦hH,⌘⌦hH).

(c) Ifo satisfies (d) and 23⁺ satisfies (a), there exists a non-zero H-intertwiner 8_o,˜ :1⌦17!1⌦Eo(˜):( ₁)_o⌦hH! o⌦hH.

(8_o,˜), for 23⁺ satisfying (a), is a basis ofHom_H(( ₁)_o⌦hH, _o⌦hH).

We note that ₁(cw_˜0) ¹⌦T_{˜ ˜w0} 2⌘⌦hHdoes not depend on the choice of the lift w˜₀ ofw₀2Y . We set

3 := { 23| = }, resp.3⁺:=3⁺\3 . (12) The idempotent e := |Z_k| ¹P

t2Zk (t) ¹t of R[Z_k] is central in R[3 (1)], and the R-linear map

⌦R[Zk]R[3 (1)]!e R[3 (1)] 1⌦˜ 7!e ˜ ( 23 ) (13) is an isomomorphism. Any R-algebra A with a basis(a_˜) ₂₃+ satisfying

a_˜a_˜0 = (t)a_˜00 for , ⁰, ⁰⁰23⁺,t 2Z_k,˜ ˜⁰=t˜⁰⁰, (14) is canonically isomorphic to the algebrae R[3⁺(1)]with its natural basis(e ˜) ₂₃+.

For an orientation o, the R-submodule A⁺o, of basis (Eo(˜))_˜23+(1) is a subalgebra of H. The algebra ⌦R[Zk]A⁺o, of basis (1⌦E_o(˜)) ₂₃+ is an R-algebra with a basis satisfying (14).

A spherical Hecke algebra is the algebra ofH-intertwiners of a rightH-module⌘⌦hH induced from a character⌘ofh, by analogy with the reductive p-adic groups

H(h,⌘):=End_H(⌘⌦hH).

Theorem1.2with⌘₁=⌘becomes the following.

Theorem 1.3(A Satake-type isomorphism for the spherical algebra). (a) A basis of the spherical Hecke algebraH(h,⌘)is(8_˜) ₂₃+, where

8_˜ :1⌦17!1⌦E_˜ :⌘⌦hH!⌘⌦hH, E_˜ := X

w₀2Y

(cw_˜0)⌦T_{˜ ˜w0},

Y = {w₀2 hS S⌘i| ^w0 = ,`( w<sub>0</sub>)=`( ) `(w₀)}.

(b) Let o be an orientation such that ⌘= o. For 23⁺, there exists an injective h-intertwiner

8_o,˜ :1⌦17!1⌦E_o(˜):⌘⌦hH!⌘⌦hH.

(8_o,˜)₂₃+is a basis of the spherical Hecke algebraH(h,⌘)satisfying (14), inducing an isomorphism

H(h,⌘)'e R[3⁺(1)].

We suppose now that 3_T exists. The center Z of H is the algebra A^Wo⁽¹⁾ of W(1)-invariants ofAo, and is a free R-module of basis

E(C)˜ =X

˜2C˜

E_o(˜) (15)

(E(C)˜ is independent of the choice of o) for all finite conjugacy classes C˜ of W(1).

We denote by C(µ)˜ the W(1)-conjugacy class of µ˜ for µ23⁺_T. The R-subspace of

basis (E(C(µ)))˜ _µ23+

T is a central subalgebra ZT of H which has better properties thanZ.

A central elementx2Z induces naturally aH-intertwiner of ⌘⌦hH:

8_x :1⌦h 7!1⌦xh=1⌦hx forh2H. (16) It is straightforward to check that 8_x belongs to the center Z(⌘,h) of H(⌘,h). The R-subspace of basis (8_E(C(µ))˜ )_µ23+

T is a central subalgebra Z(⌘,H)_T of the spherical algebra H(⌘,h).

Theorem 1.4 (Almost-isomorphism between the centers ofHandH(⌘,h)). We suppose that 3_T exists. Let⌘ be a character ofh.

(a) ZT is a finitely generated central R-subalgebra ofH, and His a finitely generated ZT-module. This is also true for(Z(⌘,H)_T,H(⌘,h))instead of(ZT,H).

(b) 8_E(C(µ))˜ =8_o,µ˜ forµ23⁺_T and any orientationo such that ⌘= o.

The linear mapµ˜ 7!8_E(C(µ))˜ : R[ ˜3⁺_T]!Z(⌘,H)_T is an algebra isomorphism.

(c) The mapx7!8_x :Z!Z(⌘,H)restricts to an isomorphismZT !Z(⌘,H)_T. We prove (a) over any commutative ring R.

We transfer these results to the groupG. The spherical Hecke algebraHR(G,K,⇢)= End_RGc-Ind^G_K⇢ of an irreducible smooth representation⇢ of K withHR(K,I(1))acting by⌘on⇢^I(1)is isomorphic toH(⌘,h)by the pro-p-Iwahori invariant functor. We denote by ZR(G,K,⇢)_T the algebra corresponding to Z(⌘,H)_T. We denote by HR(Z⁺,Z₀, ) the R-algebra of elements in the Hecke algebraHR(Z⁺,Z₀, )with support contained in the monoid Z⁺ ofz2Z with⌫_Z(z)dominant.

From Theorem1.3we obtain an algebra isomomorphism

So:HR(G,K,⇢)!HR(Z⁺,Z0, ) (17) for each orientationo such that⌘= o. This isomorphism restricts to an isomorphism, independent of the choice ofo,

ST :ZR(G,K,⇢)_T !HR(T⁺,T₀, ). (18) Let ⇡ be a smooth R-representation of G such that HomR(⇢,⇡) contains a ZR(G,K,⇢)_T-eigenvector A of eigenvalue ⇠, seen as an homomorphism 3˜⁺_T ! R (Theorem1.4). From Theorem 1.4, forv2⇢^I(1) non-zero andµ23⁺_T,

⇠(µ)A(v)˜ =A(v)E_o(µ)˜ =A(v)E(C˜(µ)).

Theorem 1.5 (Supersingularity in G and in H). The eigenvalue ⇠ of the ZR(G,K,⇢)_T-eigenvector A2Hom_R(⇢,⇡) is supersingular if and only if the submodule

A(v)Hof ⇡^I(1) is supersingular.

We recall that an homomorphism 3˜⁺_T ! R is called supersingular if it vanishes on the non-invertible elements, and that a simple right H-module M is called supersingular ifM E(C)˜ =0for all finite conjugacy classesC˜ inW(1)with positive length [13, Definition 1]. This is equivalent to M E(C(µ))˜ =0for all non-invertibleµ23˜⁺_T.

In a forthcoming article, we will study the parabolic induction forH-modules; we hope to prove that the isomorphismSo(17) is the Satake isomorphism of [7] for a good choice ofosuch that⌘= o(this was proved by Ollivier [10, Theorem 5.5]), whenGis split with a simply connected derived group, and K is hyperspecial; as Z =T, we have So=ST, and that an irreducible smooth admissible representation ⇡ is supersingular if and only if⇡^I(1) contains a supersingular module (this was proved by Ollivier forG=G L(n,F) and PG L(n,F)[11, Theorem 5.26]).

Finally, we classify the supersingular simple finite-dimensionalH-modules (proved by Ollivier whenG is split, andK is hyperspecial [11, Corollary 5.15]).

For a character 4 of H^aff, the R-subalgebra H⁴ of H generated by H^aff and the

(1)-fixator of4,

(1)₄:= {u 2(1)|4(uhu ¹)=4(h)forh2H^aff},

is identified by (3) with the twisted tensor product H^aff⌦R[Zk]R[(1)₄]!H⁴. For a simple finite-dimensional R-representation of(1)4 equal to4 on Z_k, let

M(4, ):=(4⌦ )⌦_H4H (19)

be the rightH-module induced from the right H⁴-module4⌦ . The induced module M(4, ) is finite dimensional. Two pairs (4₁, ₁), (4₂, ₂) are called conjugate by an elementu 2(1)if

4₁(uhu ¹)=4₂(h), ₁(uvu ¹)= 2(v) for(h, v)2H^aff⇥u ¹₄(1)u.

The affine Coxeter system(W^aff,S^aff)is the direct product of the irreducible affine Coxeter systems (W_i^aff,S_i^aff)_16i6r associated to the irreducible components(6_i,1_i)_16i6r of the based reduced root system(6,1). TheR-submodule of basis(Tw_˜)_w˜

i2W_i^aff(1)is a subalgebra H^aff_i ofH^aff. The algebrasH^aff_i are called the irreducible components ofH^aff.

Theorem 1.6(Supersingular simple modules). (a) The characters 4 of H^aff are in bijection with the pairs( ,J), where 2 Zˆ_k and J⇢S^aff, =4|Zk, and J =S₄^aff (10). When S₄^aff=S^aff,4 is called a sign character, and the character 4 ◆ (4) is called a trivial character.

(b) A character 4 of H^aff is supersingular if and only if it is not a sign or trivial character on each irreducible component ofH^aff.

(c) A finite-dimensional rightH-module is supersingular if and only if it is isomorphic to M(4, ), where 4 is a supersingular character of H^aff and is a simple finite-dimensional R-representation of (1)₄ equal to 4 on Zk.

(d) M(4₁, ₁)'M(4₂, ₂)if and only if(4₁, ₁), (4₂, ₂)are(1)-conjugate.

2. The characters of h and H^aff

Proposition 2.1. A simpleh-module has dimension 1.

Proof. The finite-dimensional R-algebra h is generated by Zk and T_s˜ for all s2 S. By the hypothesis on R (§1.4), a right simpleh-module is finite dimensional and contains an eigenvectorv of Z_k. Following the argument of [4, Theorem 6.10], we choose win the finite groupW0of maximal length such thatvTw_˜ 6=0, and we show thatRvTw_˜ ish-stable.

RvTw_˜ is stable byTt, because Tw_˜Tt =(w•t)Tw_˜ fort 2Zk. RvTw_˜ is stable byT_s˜, because

– if`(ws)=`(w)+1,vTw_˜T_s˜=vTws˜ and by the hypothesis onw,vT_w˜s˜=0;

– if `(ws)=`(w) 1, Tw_˜T_s˜=T_w˜s˜ 1T_s˜²=T_w˜s˜ 1c_s˜T_s˜=T_ws˜ 1T_s˜c_s˜=(w•c_s˜)Tw_˜. We used that T_˜s andc_s˜ commute.

Proposition 2.2. The characters⌘ofhare in bijection with the pairs( ,J), where 2 Zˆk

and J⇢S (9), by the recipe

⌘|Zk = , S⌘= {s2 S|⌘(T_s˜)6=0} =J.

We have ⌘(T_s˜)= (c_s˜)if s2 J.

The characters 4 of H^aff are in bijection with the pairs ( ,J), where 2 Zˆ_k and J⇢S^aff, by the recipe

4|Zk = , S₄^aff= {s2S^aff|4(T_s˜)6=0} = J.

We have 4(T_s˜)= (c_s˜)if s2 J.

The set J is independent of the choice of the lifts˜ofs. We call( ,J)the parameters of the character. The restriction tohof the character4ofH^aff with parameters( ,S₄^aff) is the character of parameters ( ,S₄^aff\S).

Proof. The proposition follows from the Iwahori–Matsumoto presentation in both cases.

If⌘|Zk = , we have

⌘(T_s˜)(⌘(T_s˜) (c_s˜))=0 fors2S. We can replace⌘,S by4,S^aff.

The involutive automorphism◆ofH(4) has the property fors2 S that

⌘(T_s˜)=0,⌘ ◆(T_˜s)=⌘(c_s˜).

The same holds for(4,S^aff)instead of(⌘,S).

Lemma 2.3. Let ⌘ be a character with parameters( ,S⌘)ofh. Then⌘ ◆ is a character of hwith parameters ( ,S S⌘). We can replace⌘,S,hby 4,S^aff,H^aff.

Leto be an orientation. We recall the notation (5), (6), (9), (10).

Lemma 2.4. Let ⌘ be a character of h with parameters ( ,S⌘). Then S⌘=S \S_o,

⌘(E_o(s))˜ =0 for alls2S. When this holds true, we denote⌘= o. We can replace(⌘,h,S, _o)by(4,H^aff,S^aff, _o^aff).

Proof. We compare the values of Eo(s)˜ and⌘(T_s˜)fors2 S:

E_o(s)˜ =T_s˜,s2S S_o,

=T_s˜ c_s˜,s2So,

⌘(T_s˜)=0,s2 S S⌘,

= (c_˜s)6=0 ifs2 S⌘. We see that

ifs2S S , then ⌘(Eo(s˜))=⌘(T_s˜)= (c_s˜)=0;

ifs2S S⌘, then ⌘(Eo(s))˜ =⌘(T_s˜)=0,s62(S S⌘)\So; ifs2S⌘, then⌘(E_o(s))˜ =0,s2S⌘\S_o.

Hence we obtain the lemma for⌘. The proof is the same for4.

Example 2.5. For the dominant orientationo⁺, S_o^aff₊=S, and the parameters of _o+ and of _o^aff₊ are( ,S ).

For the anti-dominant orientation o , S^aff_o =S^aff S, and the parameters of _o are ( ,;), while those of _o^aff are( ,S^aff S ).

Lemma 2.6. (i)Any subset of S is equal to S_o for some orientationo.

A character⌘of hof restriction to Z_k is equal to _o for some orientationo, and

⌘= o,So\S =S⌘.

(ii) Two R-characters ⌘₁,⌘ of h of parameters ( ₁,S⌘₁), ( ,S⌘)are equal to ( ₁)_o, _o for some orientation o if and only if

S⌘₁\S =S⌘\S ₁.

In this case, ⌘₁=( ₁)_oand⌘= o,S_o\(S ₁[S )=S⌘₁[S⌘.

Proof. (i) Letw_o2W0. For↵21, the root in {↵, ↵}positive onw_o¹(D⁺)is equal to

↵_o=↵ifw_o(↵) >0and↵_o= ↵ifw_o(↵) <0; hence s↵ 2S_o,w_o(↵) >0.

For a subset X of S, we have X =So for the orientation o=o⁺•w_o of Weyl chamber D_o=w_o¹(D⁺), wherew_o is the longest element of the grouphS Xi(w=1 ifS=X).

(ii) So\S ₁ =S⌘₁ and So\S =S⌘ imply that So\S ₁\S =S⌘₁\S =S⌘\S ₁. If S⌘₁\S =S⌘\S ₁, then So\(S ₁[S )=S⌘₁[S⌘ implies that So\S ₁ =S⌘₁ and So\ S =S⌘.

Definition 2.7. A character ofhnot vanishing onT_s˜ for alls2 S is called a twisted sign character, and its image by the involution◆ is called a twisted trivial character.

We make the same definition forH^aff,S^aff replacingh,S.

The twisted sign characters ⌘ are never 0 on Tw_˜ for w2W0. The algebra h admits no twisted sign or trivial characters whenc_s˜=0 for somes2S. They are equal to _o+, where 2Zˆ_k satisfies S =S.

The twisted trivial characters ⌘vanish on Tw_˜ for all w2W0. They are equal to _o , where 2Zˆk satisfies S =S.

The same remarks can be made forH^aff, (W^aff,S^aff)replacingh, (W₀,S).

3. Distinguished representatives of W0\W

We recall a well-known lemma for the affine Coxeter system (W^aff,S^aff)extended to the group W =W^affo.

Fors2 S^aff, we denote by A_s the unique positive affine root such thats(A_s)is negative.

We haves(A_s)= A_s [8, 1.3.11]. Whens2S we write A_s =↵_s. Lemma 3.1. (1) For(s, w)2S^aff⇥W, we have

`(ws)=1+`(w),w(↵_s) >0.

(2) Forv6w in W ands2 S^aff, we have (a) eithersv6wor sv6sw;

(b) eitherv6swor sv6sw.

Proof. We recall that W =W^affo. Let(s,u, w)2 S^aff⇥⇥W^aff.

(1) We have `(uws)=`(ws),`(uw)=`(w), and `(ws)=`(w)+1,w(↵_s) >0 [8, 1.13.13]. By definition (§1.2) an affine root is positive if and only if it is positive on the dominant alcoveC⁺. As the group normalizes C⁺, it normalizes the set of positive affine roots, in particularw(↵_s) >0,(uw)(↵_s) >0.

(2) Let (v,u⁰)2W^aff⇥. By definition of the Bruhat–Chevalley partial order [14, Ap. 2],vu⁰6wu is equivalent tou⁰=u, v6w. InW^aff [8, 1.3.19],

(a) eithersv6wor sv6sw;

(b) eitherv6swor sv6sw.

We multiply (a) and (b) byu on the right without changing6.

Remark 3.2. As`(w)=`(w ¹)andv6w,v ¹6w ¹, in Lemma3.1(1) we also have

`(sw)=1+`(w),w ¹(↵_s) >0, and in Lemma3.1(2), (a) and (b) can be replaced by (c) eithervs6w orvs6ws;

(d) eitherv6ws orvs6ws.

We introduce now a distinguished setDof representatives of W0\W. Proposition 3.3. The three sets

D1= {d 2W |d ¹(↵) >0 for all↵26⁺},

D2= { w₀|( , w₀)23⁺⇥W0,`( w<sub>0</sub>)=`( ) `(w<sub>0</sub>)}, D3= {d 2W |`(w₀d)=`(w<sub>0</sub>)+`(d)for allw₀2W₀},

are equal, and will be denoted byD. The cosets W0d, ford 2D, are disjoint of union W. Proof. The setD1is also equal to

{d 2W |`(sd)=`(d)+1 for alls2 S}, (20) because one can restrict to ↵21 in the definition of D1 and, for s2 S, d ¹(↵_s) >

0,`(sd)=`(d)+1 (Remark 3.2). Let w2W not in D1. There exists s2S with

`(sw)=`(w) 1. Then w₁=sw satisfies `(w)=1+`(w₁). We reiterate, and after finitely many steps we obtain(w₀,d)2W₀⇥D1such thatw=w₀d,`(w)=`(w₀)+`(d).

The pair(w₀,d)is unique. Indeed, ford,d⁰inD1withd⁰d ¹2W₀, for all↵21we have d⁰d ¹(↵)= 26, and d ¹(↵)=d^{0 1}( ) is positive asd 2D1; hence >0 as d⁰2D1. This impliesd =d⁰. We deduce thatD1is a set of representatives ofW₀\W, thatd 2D1

is the unique element of minimal length in W₀d, and that D1⇢D3. This implies that D1=D3.

We now compare the setsD1andD2. For( , w₀)23⇥W₀, we deduce from Lemma3.1 (see [15, Corollary 5.11]) that

`( w<sub>0</sub>)=`( ) `(w₀),↵ ⌫( ) >0 for all↵26⁺\w₀(6 ). (21) On the other hand, for all↵26⁺,( w₀) ¹(↵)=w₀¹(↵)+↵ ⌫( )is positive if and only if

w₀¹(↵) >0,↵ ⌫( )>0 or w₀¹(↵) <0,↵ ⌫( ) >0 (22) [15, (36)]. Comparing (21) and (22), we deduce thatD1=D2.

Remark 3.4. (i) The distinguished set D^aff of representatives of W₀\W^aff given by Proposition3.3applied toW^aff is equal toD^aff=D\W^aff, andD=D^aff.

(ii) The distinguished set D of representatives of W₀\W^aff can be inductively constructed: it is the set of w₀2D for 23⁺ and w₀2W0, such that w₀=1 orw₀has a reduced decomposition w₀=s₁. . .s_r (s_i 2S), such that

`( s1. . .si+1)=`( s1. . .si) 1 for16i 6r.

Note that s2D,↵_s ⌫( ) >0whens2S.

We denote byw₁ the unique element of maximal length in the finite Weyl groupW0. Lemma 3.5. Let , µ23⁺. The double W₀-coset W₀ W₀ has a unique element w of maximal length,

w =w₁ , `(w )=`(w₁)+`( ) and 6µ,w 6w<sub>µ</sub>. The setW<sub>0</sub> W<sub>0</sub>\Dis equal to D( )= { w<sub>0</sub>|w<sub>0</sub>2W<sub>0</sub>,`( w₀)=`( ) `(w₀)}.

Proof. The coset W0d ofd 2Dcontains a unique element of maximal length, equal to w₁d,`(w<sub>1</sub>d)=`(w₁)+`(d). For 23<sup>+</sup>, the set D\W0 W0 contains a unique element of maximal length, equal to (Remark3.4(ii)). HenceW<sub>0</sub> W<sub>0</sub>contains a unique element w of maximal length, equal to w<sub>1</sub> and `(w )=`(w<sub>1</sub>)+`( ). As wµ=w₁µ,`(wµ)=

`(w<sub>1</sub>)+`(µ), the equivalence 6µ,w₁ 6w₁µ is clear. We have D( )= W0\D (Proposition 3.3), and µ2W₀ W₀,µ=w w ¹ for some w2W₀,µ= , as 3⁺ represents the orbits ofW₀in 3[7, 6.3].

Lemma 3.6. Let ( , w₀)23⁺⇥W₀,d = w₀2D, and letµ23⁺. (1) Fors2 S^aff,ds62D,dsd ¹2 S)`(ds)=`(d)+1.

(2) Fors2 S andds 2D, we have `(ds)=`(d)+1,`(w<sub>0</sub>s)=`(w₀) 1.

(3) For(w,d⁰)2W0⇥D, we have d 6wd⁰)d 6d⁰. (4) Fors2 S such that ds2D, we haved 6µ)ds6µ.

(5) We haved 6wµ,d 6µ, 6µ.

Proof. (1) Lets2S^aff. By (20) and Remark3.2,

ds62D,(ds) ¹(↵) <0 for some ↵21.

Asd ¹( ) >0for all 26⁺, anddsd ¹2W^aff, we have

s((d ¹(↵))) <0,d ¹(↵)= As ,↵=d(As),s↵ =dsd ¹. We have`(ds)=`(d)+1 by Lemma3.1(1).

(2) Lets2 S withds2D. Then

`(ds)=`(d)+1,`( ) `(w₀s)=`( ) `(w₀)+1,`(w<sub>0</sub>s)=`(w₀) 1.

(3)d 6wd⁰ ands2 Simply thatd 6swd⁰orsd 6swd⁰ by Lemma3.1(2); as d<sd, we obtain

d6wd⁰)d 6swd⁰.

Ifw6=1, we chooses such that sw < w. Repeating the procedure, we obtaind6d⁰ by induction on the length ofw2W₀.

(4) Asd 6µ,ds 6µ ords 6µsby Lemma 3.1(2). Whenµs< µ, we obtainds6µ.

Suppose thatµs> µandds 6µs. By Lemma3.1(1),

`(µs)=`(mu)+1,µ(↵_s)=↵_s ↵_s ⌫(µ) >0,↵_s ⌫(µ)60, ,↵_s ⌫(µ)=0,⌫(µ)fixed bys,µs=sµu,u23\.

We deduce thatds 6sµu. By (3),ds6µu, becauseds, µu2D. As3 is commutative, ds 6uµ. For w2W, there is a unique element uw2 such that w2uwW^aff. By the definition of the Bruhat–Chevalley order,d 6µ,ds 6uµimply thatud =uµ=uuµ. We deduce thatu =1,ds 6µ.

(5) The implications d6wµ(d 6µ( 6µ are obvious, becaused 6 , µ6wµ. The implicationd 6w_µ)d 6µfollows from (3), becausew_µ=w₁µ(Lemma3.5) and µ2D. The implicationd 6µ) 6µ follows from (4) reiterated finitely many times fors2S such that`(ds)=`(d)+1 ifd 6= (Remark3.4(ii)).

Remark 3.7. Results similar to Proposition 3.3 and Lemma 3.6 are already in [9, Proposition 2.5, Lemma 2.6, Proposition 2.7], [10, Lemma 2.4], [11, Proposition 1.3], when W is the Iwahori Weyl group of a split reductive p-adic groupG.

Lemma 3.8. In Lemma 3.6, for s2S and1 as in (11),

ds62D,dsd ¹=w₀sw₀¹2S ,w₀(↵_s)21 ,w₀(↵_s)26⁺, w₀(↵_s) ⌫( )=0. This implies that`(w<sub>0</sub>s)=`(w₀)+1 and`(ds)=`(d)+1=`( ) `(w₀s)+2.

Proof. By Lemma 3.6(1), ds62D,d(↵s)= w₀(↵_s)=w₀(↵_s) w₀(↵_s) ⌫( )21, w₀(↵_s)21, w₀(↵_s) ⌫( )=0,w₀(↵_s)21 . In the proof of Lemma 3.6(1), we saw thatdsd ¹=sw₀(↵_s)=w₀sw₀¹. Note thatds62Dimplies that`(ds)=`(d)+1=`( )

`(w<sub>0</sub>)+16=`( ) `(w<sub>0</sub>s). Hence `(w₀s)=`(w<sub>0</sub>)+1,`(ds)=`( w<sub>0</sub>s)=`( ) `(w₀s)+ 2.

By (22),ds2D,↵ ⌫( ) >0 for all↵26⁺\w₀s(6 ). We have 6⁺\w₀s(6 )=(6⁺\w₀(6 )) {w₀( ↵_s)} ifw₀(↵_s)26 ,

=(6⁺\w₀(6 ))[{w₀(↵_s)} ifw₀(↵_s)26⁺,

because, for 26⁺, we have sw₀¹( ) <0 if and only if 2{w₀(↵_s)}[(w₀(6 ) {w₀( ↵_s)}), as recalled at the beginning of this section. As d 2D, we have↵ ⌫( ) >0 for all↵26⁺\w₀(6 ). We deduce thatds62D,w₀(↵_s)26⁺, w₀(↵_s) ⌫( )=0.

4. h-eigenspace in⌘⌦hH

Proposition 4.1. For any choice of lift d˜of d2DinD(1), the lefth-moduleHis free of basis(T_d˜)_d2D, and the right h-module His free of basis(T_d˜ 1)_d2D.

Proof. To the set D of distinguished representatives of the right W0-cosets in W is associated a disjoint union W(1)=F

d2DW₀(1)d. Hence˜ Hadmits the R-bases (T_wd˜)_w2W0(1),d2D and (T_d˜ 1w)_w2W0(1),d2D.

A basis of h is (Tw)_w2W0(1). By the braid relations, T_wd˜=TwT_d˜ and T_d˜ 1w =T_d˜ 1Tw, because `(wd)=`(w)+`(d).

Remark 4.2. An element ofHcan be written as a sum P

d2Dh_d˜T_d˜, whereh_d˜2h, and, fort 2Zk,

h_d˜T_d˜=h_td˜T_td˜=h_td˜htT_d˜, h_d˜=h_td˜ht.

The monoid3⁺ represents the orbits of W0 in 3, and the double(W0,W0)-cosets of W, because W =3oW0. The(h,h)-moduleHis the direct sum

H= M

23⁺

h( ) (23)

of the(h,h)-submodulesh( )of R-basis(Tw)_{w2W0(1)˜W0(1)}. We setD( ):=W₀ W₀\D.