The pro-p-Iwahori Hecke algebra of a reductive p-adic group I

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The pro-p-Iwahori Hecke algebra of a reductive p-adic group I

Vigneras Marie-France July 10, 2015

version 4

Abstract

Let R be a commutative ring, let F be a locally compact non-archimedean field of finite residual fieldkof characteristicp, and letGbe a connected reductiveF-group. We show that the pro-p-Iwahori HeckeR-algebra ofG=G(F) admits a presentation similar to the Iwahori-Matsumoto presentation of the Iwahori Hecke algebra of a Chevalley group, and alcove walk bases satisfying Bernstein relations. This was known only for a F-split group G.

1 Introduction

This paper extends to a general reductive p-adic group Gthe description of the pro-p- Iwahori Hecke algebra over any commutative ring R, that I gave 10 years ago for a split group G. This is a basic work which allows to describe the center and to prove finiteness results of the pro-p-Iwahori Hecke algebra for any (G, R). It is a fundamental tool for the theory of the representations of Gover a field C of characteristic p: the inverse Satake isomorphism for spherical Hecke algebras, the classification of the supersingular simple modules of the pro-p-Iwahori Hecke C-algebra, and the classification of the irreducible admissible smooth C-representations ofGin term of parabolic induction and irreducible supersingular representations of the Levi subgroups.

The study of congruences between classical modular forms naturally leads to representations over arbitrary commutative rings R, rather than to complex representations.

In our local setting, that means studyingR-modules with a smooth action ofG=G(F) where F is a locally compact non-archimedean field of finite residue field k, and Gis a connected reductive group overF.

WhenCis an algebraically closed field of characteristic equal to the characteristicpof k, very little is known about the theory of the smoothC-representations ofG, besides the basic property that a non-zero representation has a non-zero vector invariant by a pro- p-Iwahori subgroup. The pro-p-Iwahori subgroups ofGare the analogues of thep-Sylow subgroups of a finite group and the study of the smoothC-representations ofGinvolves naturally the pro-p-Iwahori HeckeC-algebraHC(1) ofG. This is our motivation to study the pro-p-Iwahori Hecke algebra ofG.

Forany triple (R, F,G), we show that the pro-p-Iwahori HeckeR-algebra HR(1) of G admits a presentation, generalizing the Iwahori and Matsumoto presentation of the Iwahori Hecke R-algebra of a Chevalley group. The proof of the quadratic relations is done by reduction to the analog HeckeR-algebra of a finite reductive group. The Iwahori HeckeR-algebraHR ofGis a quotient ofHR(1) and all our results transfer to analogous and simpler results for the Iwahori HeckeR-algebra.

The Iwahori-Matsumoto presentation of the pro-p-Iwahori HeckeR-algebra ofGleads naturally to the definition of R-algebras HR(qs, cs) associated to a group W(1) and parameters (qs, cs) satisfying simple conditions. The groupW(1) is an extension by a commutative groupZkof an extended affine Weyl groupW attached to a reduced root system Σ. The groupW is more general than the group appearing in the Lusztig affine Hecke al- gebrasHR(qs, qs−1). TheR-algebraHR(qs, cs) is a freeR-module of basis indexed by the elements of W(1) satisfying the braid relations and quadratic relations with coefficients (qs, cs).

We show that the algebraHR(qs, cs) admits, for any Weyl chamber, an alcove walk basis indexed by the elements ofW(1), a product formula involving alcove walk bases associated to different Weyl chambers, and Bernstein relations. When the qs are invertible inRwe obtain a presentation of the algebraHR(q_s, c_s) generalizing the Bernstein-Lusztig presentation for the Iwahori Hecke algebra of a split group. Our proofs proceed by reduction to the case q_s= 1.

We recall that we put no restriction on the triple (R, F,G), the reductive group G may be not split, the local field may have characteristicp, the commutative ringR may be the ring of integers Zor a field of characteristicp.

WhenGis split, the complex Iwahori Hecke algebraH_CofGwas well understood. It is the affine Hecke algebra attached to a based root datum ofGand to the cardinalqofk (the first proof is due to Iwahori and Matsumoto for a Chevalley group). Starting from the Iwahori-Matsumoto presentation ofH_C, Bernstein and Lusztig gave another presentation of H_C, from which one can recover the center of H_C and which is an essential step for the classification of its simple modules. The classification was done for G = GL(n) by

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Zelevinski and Rogawski, and forGsimple, simply connected with a connected center by Kazhdan-Lusztig, and Ginzburg, using equivariant K-theory of the variety of Steinberg triples. The condition “simply connected” was waived by Reeder. G¨ortz realized that the Bernstein basis could be understood using Ram’s alcove walks, and gave a simpler proof of the Bernstein presentation of an affine Hecke complex algebra of a based root datum with unequal invertible parameters. WhenGis split but for any pair (R, F), I had shown that the pro-p-Iwahori Hecke algebra ofGadmits an Iwahori-Matsumoto presentation and an integral Bernstein basis, using Haines minimal expressions. A student Nicolas Schmidt of Grosse-Kloenne, in his unpublished diplomarbeit, defined the alcove walk basis, proved the product formula, and studied the Bernstein relations for algebras HR(qs, cs) containing the algebras arising from a split G, but not all those arising from a generalG.

For the field of complex numbersC, the pro-p-Iwahori-invariant functor is an equiv- alence of categories from the C-representations ofGgenerated by their vectors invariant by a pro-p-Iwahori subgroup onto the category of right H_C(1)-modules. When C is replaced by an algebraically closed field C of characteristicp, this does not remain true : the functor does not always send irreducible representations onto simple modules. How- ever the pro-p-Iwahori Hecke algebraH_C(1) appears constantly in the theory of smooth C-representations ofG. The most striking example is the following:

For all integers n > 2, there exists a numerical Langlands correspondance for the pro-p-Iwahori-Hecke C-algebra H_C(1) of GL(n, F) : a bijection between the simple supersingular H_C(1)-modules of dimension n and the dimension n irreducible continuous C-representations of the Galois group Gal(F^s/F) of the separable closureF^sofF.

In a forthcoming work with Abe, Henniart and Herzig [AHHV], we classify the irreducible admissible C-representations of G which are not supercuspidal, in term of the irreducible admissible supercuspidal (= supersingular) representations of the Levi subgroups and of parabolic induction. The Bernstein relations in the pro-p-Iwahori Hecke C-algebra HC(1) of G, which is isomorphic to a C-algebra HC(0, cs) with parameters qs= 0 is one of the main ingredients of the classification.

In a sequence to this paper, we describe the center of HR(1), the inverse Satake isomorphisms and we give the classification of the simple supersingular HC(1)-modules, extending the work of myself and of Rachel Ollivier done only for split groups G.

I had many conversations with Noriyuki Abe on the Bernstein relations and on their relations with the change of weight, during his stay at the “Institut de mathematiques de Jussieu” in 2013. The unpublished diplomarbeit of Nicolas Schmidt was very helpful for writing this article. I am very thankful to an excellent referee and Florian Herzig for their careful reading.

2 Main Results

2.1 Iwahori-Masumoto presentation.

Let G be a connected reductive group over a local non-archimedean field F of finite residue fieldk of characteristicpwith qelements, and letR be a commutative ring. We fix an Iwahori subgroup I ofG. Its pro-p-radicalI(1) is called a pro-p-Iwahori group. It is the unique pro-p-Sylow subgroup of I, and of every parahoric subgroup containing I.

The pro-p-Iwahori subgroups ofGare all conjugate. The Iwahori Hecke ring H=Z[I\G/I]

with the convolution product, is isomorphic to the ring of intertwiners End_Z_[G]Z[I\G] of the regular right representationZ[I\G] ofGassociated toI. The Iwahori HeckeR-algebra

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obtained by base change

(1) H_R=R⊗_ZH=R[I\G/I].

is isomorphic to theR-algebra of intertwiners End_R[G]R[I\G]. We replaceIbyI(1) and define in the same way the pro-p-Iwahori Hecke ring H(1) = Z[I(1)\G/I(1)] and the pro-p-Iwahori HeckeR-algebraHR(1).

The sets I\G/I and I(1)\G/I(1) have a natural group structure, isomorphic to the Iwahori Weyl group W and the pro-p-Iwahori Weyl groupW(1) defined as follows.

The Iwahori groupIis the parahoric subgroup ofGfixing an alcoveCin the building of the adjoint group ofG. To defineW andW(1) we choose an apartmentAcontaining C. The apartmentA is associated to a maximalF-split subtorusT of G. Let Zand N denote the centralizer and the normalizer of Tin G, and Z :=Z(F), N := N(F) their F-rational points. Then W = N/N ∩I and W(1) =N/N ∩I(1). We check that these maximal split tori are conjugate by I. The same is true for their normalizers and for the corresponding groups W andW(1).

The apartmentAis a finite dimensional affine euclidean real space with a locally finite setHof hyperplanes, such that the orthogonal reflections with respect toH ∈Hgenerate an affine Weyl groupW(H), andCis a connected component ofA−∪_H∈HH. The groupN acts on Aby affine automorphisms respectingHand its subgroupZ acts by translations.

The parahoric subgroups of G generate a subgroup G^{af f}, which is also the kernel of the Kottwitz morphism κG, and Gis generated by Z ∪G^{af f}. The maximal compact subgroup ˜Z0 of Z acts trivially on Aand contains the unique parahoric subgroup Z0 of Z, of unique pro-p-Sylow subgroup Z0(1). The group Zo is the kernel of the Kottwitz morphism κZ of Z and the quotient Zk =Z0/Z0(1) is the group of rational points of a k-torus. We have

Z∩I=Z0, Z∩I(1) =Z0(1), I=I(1)Z0. (2)

The action ofN^{af f} =N∩G^{af f} on the apartmentAinduces an isomorphism

(3) W^{af f} =N^{af f}/Z0→W(H).

The groups Z0(1) ⊂ Z0 are normalized byN and the maps n 7→ InI, n 7→ I(1)nI(1) induce bijections

(4) W =N/Z0→I\G/I, W(1) =N/Z0(1)→I(1)\G/I(1).

The pro-p-Iwahori-Weyl group W(1) is an extension of the Iwahori-Weyl group W by Z₀/Z₀(1)'I/I(1)

(5) 1→Z_k→W(1)→W →1.

The extension does not split in general (see [VigMA]). For a subset X ⊂W, we denote byX(1) the inverse image ofX inW(1). For an elementw∈W, we denote by ˜w∈W(1) an element liftingw(hence ˜w∈w(1)).

For n ∈ N, the double coset InI depends only on the image w ∈ W of n and the corresponding intertwiner in the Iwahori Hecke ringHis denoted byTw. Thus (Tw)_w∈W is a natural basis ofH. We do the same forH(1) andW(1). The relations satisfied by the products of the basis elements follow from the fact thatW is a semi-direct product of the affine Weyl groupW^{af f} by the image Ω inW of theN-normalizer ofC,

(6) W =W^{af f} oΩ.

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The group Ω identifies with the image of the Kottwitz morphism κ_G. Let S^{af f} ⊂W^{af f} be the set of orthogonal reflections with respect to the walls of C, using the isomorphism (3). The length `of the Coxeter system (W^{af f}, S^{af f}) inflates to a length ofW constant on the double cosets modulo Ω, and to a length of W(1) constant on the double cosets modulo the inverse image Ω(1) of Ω. The Bruhat order ofW^{af f} inflates toW(1) and to W as in [Vig, Appendix].

Forn∈N of image winW or inW(1), the sets

(7) InI/I 'I(1)nI(1)/I(1)

have the same number qw of elements. The integerqwis a power of the cardinalqof the residue field of F. Whens, s⁰ ∈ S^{af f} are conjugate inW (denote s∼ s⁰),qs =qs⁰ and qw=qs₁. . . qs_n ifw=s1. . . snuwithsi∈S^{af f}, u∈Ω is a reduced decomposition.

Theorem 2.1. The Iwahori Hecke ringH is the freeZ-module with basis(Tw)_w∈W endowed with the unique ring structure satisfying

• The braid relations: TwTw⁰ =Tww⁰ ifw, w⁰∈W, `(w) +`(w⁰) =`(ww⁰).

• The quadratic relations: T_s²=qs+ (qs−1)Ts if s∈S^{af f}.

The Iwahori HeckeR-algebra HR has the same presentation over R by base change (1).

The elements in the basis (Tw)_w∈W₍₁₎ of H(1) verify the braid relations, and similar quadratic relations but the coefficient qs−1 is replaced by an element ofZ[Zk], that we define now.

Lets∈S^{af f} and letKF_s be the parahoric subgroup ofGfixing the face ofCsupported on the wall fixed bys, of codimension 1. The quotient ofKF_s by its pro-p-radicalKF_s(1) is the groupKF_s,k ofk-points of a finite reductive connected group overkof semi-simple rank 1. Let T0 be the maximal compact subgroup of the maximal split torus T of G, let T0(1) be the pro-p-Sylow subgroup of T0 and let Tk =T0/T0(1). The group Tk is a maximal split torus of K_F_s_,k of centralizer Z_k, and the root system Φ_F_s of K_F_s_,k with respect to T_k is contained in the root system Φ of G with respect toT. We denote by N_s,k (the group ofk-points of) the normalizer ofT_k inK_F_s_,k, byU_α_s_,kthe root subgroup associated to a reduced rootα_s∈Φ_F_s (we have U_2α_s_,k⊂U_α_s_,k if 2α_s∈Φ_F_s_,k), byK_F⁰

s,k

the group generated byU_α_s_,k andU_−α_s_,k, byZ_k,sthe intersectionZ_k∩K_F⁰

s,k, and byc_s the element defined by the formula:

(8) cs= (qs−1)|Zk,s|⁻¹ X

t∈Zk,s

Tt.

The groupUα_s,kis ap-Sylow subgroup ofKF_s,k, the integerqsis the number of elements ofUα_s,k and (qs−1)|Zk,s|⁻¹is an integer. Foruk ∈U_α^∗

s,k=Uα_s,k− {1}, the intersection U_−α_s_,ku_kU_−α_s_,k∩N_s,k={m_α_s(u_k)}

consists of a single element. The image of the map mα_s : U_α^∗

s,k → Ns,k is the coset mα_s(uk)Zk,s=Zk,smα_s(uk). The square mα_s(uk)² is an element of Zk,s. We choose an arbitrary element uk ∈ U_α^∗

s,k and we denote by ˜s the image of mα_s(uk) in W(1) (as in section 4.2). Such a lift ˜s of sis called admissible. The quadratic relation of Ts˜in H(1) is the same as the quadratic relation of T_˜_s in the finite Hecke algebra H(KFs,k, U_α_s_,k).

The quadratic relation inHR(K_F_s_,k, U_α_s_,k) whenRis a large field of characteristicpwas computed by Cabanes and Enguehard [CE, Prop. 6.8].

Theorem 2.2. The pro-p-Iwahori Hecke ringH(1)is the freeZ-module with basis(T_w)_w∈W₍₁₎ endowed with the unique ring structure satisfying

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• The braid relations: T_wT_w⁰ =T_ww⁰ ifw, w⁰∈W(1), `(w) +`(w⁰) =`(ww⁰).

• The quadratic relations: T_s_˜²=qsT_˜_s2+c˜sT˜s fors∈S^{af f},

wherec_˜_s=c_sif the order ofZ_k,sisq_s−1(for example ifGisF-split), and in general there are positive integersc_˜_s(t) =c_˜_s(t⁻¹)fort∈Z_k,s, constant on the cosett{xs(x)⁻¹|x∈Z_k}, of sum P

t∈Zk,sc˜s(t) =qs−1 such that cs˜=P

t∈Zk,sc˜s(t)Tt and c_˜_s≡c_s modulo p.

The Z-module of basis (T_w)_w∈Ω(1) is a subalgebra of H(1) isomorphic to the group algebra Z[Ω(1)] by the braid relations. The Z-module of basis (T_w)_w∈Waf f(1) for the inverse imageW^{af f}(1) ofW^{af f} inW(1), is a subalgebraH^{af f}(1) andH(1) is isomorphic to the twisted product

H(1)' H^{af f}(1)⊗_Z_[Z_k_]Z[Ω(1)].

The pro-p-Iwahori Hecke R-algebra HR(1) has the same presentation by base change.

Theorems 2.1 and 2.2 imply (this can also proved directly):

Corollary 2.3. The surjectiveR-linear mapHR(1)→ HR : Tw˜ 7→Tw forw˜∈W(1)of imagew∈W, is an R-algebra homomorphism.

The properties of the pro-p-Iwahori Hecke R-algebra HR(1) are transported to the Iwahori Hecke R-algebraHR via this surjectiveR-algebra homomorphism.

We describe conditions on elements (q_s, c_s)∈R×R[Z_k] fors∈S^{af f}(1), the inverse image ofS^{af f} inW(1), implying the existence of anR-algebraH_R(q_s, c_s) of basis (T_w)_w∈W₍₁₎ satisfying braid and quadratic relations as in Thm. 2.2. We write cs=P

t∈Zkcs(t)twith cs(t)∈R.

Theorem 2.4. Let(qs, cs)∈R×R[Zk]fors∈S^{af f}(1). The following property A implies the property B :

A For allt∈Zk, s∈S^{af f}(1), w∈W(1) such thatwsw⁻¹∈S^{af f}(1), 1) qs=qst=q_wsw⁻¹.

2) cst=cst, andcwsw⁻¹ =P

t∈Zkcs(t)wtw⁻¹.

B The freeR-moduleHR(qs, cs)of basis(Tw)_w∈W(1) has a uniqueR-algebra structure satisfying

• The braid relations: TwTw⁰ =Tww⁰ ifw, w⁰∈W(1), `(w) +`(w⁰) =`(ww⁰).

• The quadratic relations: T_s²=q_ss²+c_sT_s for s∈S^{af f}(1).

InBthe braid relations imply that the group algebraR[Z_k] embeds inH_R(q_s, c_s) by the linear map sendingt∈Z_k toT_t.

If theqs are not zero divisors inR, the properties A and B are equivalent. The maps s 7→qs from S^{af f}(1) to R satisfying A1) are naturally in bijection with the maps from S^{af f}/∼toR.

For indeterminates (qs)_s∈Saf f/∼, we have the genericR[(qs)]-algebraH_R[(q_s_)](qs, cs).

TheR-algebraHR(qs, cs) is a specialisation of the generic algebra.

Proposition 2.5. Let (qs)_s∈Saf f/∼ be indeterminates, qs=q²_s, and let (cs)_s∈Saf f(1) be elements ofR satisfying A.2).

TheR[(q_s,q⁻¹_s )]-algebraH_R[(q

s,q⁻¹_s )](1,q⁻¹_s c_s)is isomorphic toH_R[(q

s,q⁻¹_s )](q_s, c_s).

The generic algebra is aR[(q_s)]-subalgebra ofH_R[(q

s,q⁻¹_s )](q_s, c_s). Different properties of the R-algebra H_R(q_s, c_s), hence of the pro-p-Iwahori Hecke algebra, are proved by reduction to the simpler caseq_s= 1 for alls∈S^{af f}/∼, using this proposition.

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Remark 2.6. The presentation of H (Thm. 2.1) generalizing the Iwahori-Matsumoto presentation for a Chevalley group [IM] cannot be found in the litterature for a general reductive groupGbut follows from different results of Bruhat and Tits [BT2, 5.2.12 Prop.

(i) and (ii)] and exercises in Bourbaki [Bki, Ch. IV,§2, Ex. 8, 22, 23, 24, 25]. Borel [Borel]

considered a semi-simple group G and a “non-connected” Iwahori subgroup ˜I = IZ˜0. When G is F-quasisplit and Z0 = ˜Z0, Z is a torus, I = ˜I, then H is a Lusztig affine Hecke algebra [Lusztig] attached to a based root datum ofGand to a system of unequal parameters (qs).

It is possible to compute the quadratic relations using the Bruhat-Tits theory without reduction to finite reductive groups (whenGisF-split [VigMA].)

2.2 Alcove walk bases and Bernstein presentation

We chose an alcoveCto defineIand an apartmentAcontainingC. To define a new basis of the algebra HR(qs, cs), we choose first a special vertexx0 ofC. The special vertices of C may be not conjugate by an element of G. The orthogonal reflections with respect of the walls containing x0 generate a group isomorphic to W0. The Iwahori-Weyl groupW is the semi-direct product

(9) W = ΛoW0 where Λ =Z/Z0.

This implies

W(1) = Λ(1)W0(1) where Λ(1) =Z/Z0(1), Λ(1)∩W0(1) =Zk. The new basis is related to this decomposition.

We identify the apartment A with an euclidean real vector space V, the vertex x₀ becoming the null vector 0 of V. We recall the natural bijections between:

the (open) Weyl chambers ofAof vertexx0, the alcoves ofAof vertexx0,

the bases of the root system Φ ofTinG,

the (spherical) orientations of (A,H) (see section 5.2).

A Weyl chamber D of V contains a unique alcove CD of vertex 0, the basis ∆D of Φ consists of the reduced roots α positive on D such that Kerα is a wall of CD. The orientation oD is such that the oD-positive side of an hyperplane H ∈ H is the set of x∈V withα(x) +r >0 whereH = Ker(α+r) forα∈Φ positive onD.

The simply transitive action ofW0 on the Weyl chambers of V inflates to an action ofW and an action ofW(1) on the spherical orientations oof (V,H), trivial on Λ(1). We denote o•w=w⁻¹(o). If Do denotes the Weyl chamber defining the orientationo, then w⁻¹(D_o) =D_o•w.

To define the new basis, we choose an orientation o of (A,H). For a pair (w, s) ∈ W^{af f}×S^{af f} such that`(ws) =`(w) + 1 we set

o(w, s) = 1 ifw(C) is contained in theo-negative side ofw(Hs),

whereHsis the affine hyperplane ofV fixed bys. Otherwise we seto(w, s) =−1. When we walk fromw(C) tows(C) we cross the hyperplaneHsin theo(w, s)-direction: positive direction ifo(w, s) = 1 and negative direction ifo(w, s) =−1.

For w ∈ W^{af f} of reduced decomposition w = s1. . . sr, si ∈ S^{af f}, r = `(w), walk- ing in a minimal gallery of alcoves C, s1(C), s1s2(C), . . . , w(C), we cross the hyperplanes Hs₁, s1(Hs₂), . . . , s1. . . sr−1(Hs_r) in theo(1, s1), o(s1, s2), . . . , o(s1. . . sr−1, sr)-directions.

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For (w, s) inW(1)×S^{af f}(1) with`(ws) =`(w) + 1, lifting an element (w^{af f}u, s) in W ×S^{af f}, withw^{af f} ∈W^{af f}, u∈Ω, s∈S^{af f}, we write

_o(w, s) :=_o(w^{af f}, s)

T_s^o^(w,s):=Ts ifo(w, s) = 1 andT_s^o^(w,s):=Ts−csifo(w, s) =−1.

We recall the quadratic relation (T_s−c_s)T_s = q_ss²; it is easy to check that c_s and T_s commute.

Theorem 2.7. Leto be a spherical orientation and letw=s₁. . . s_ruwithu∈Ω(1)and s_i ∈S^{af f}(1)for1≤i≤r=`(w). The element ofH_R(q_s, c_s)defined by

(10) Eo(w) =T_s₁ô^(1,s¹⁾T_s₂ô^(s¹^,s²⁾. . . T_s_rô^(s¹^...s^r−1^,s^r⁾Tu

does not depend on the choice of the reduced decomposition of w, satisfies (11) Eo(w)−Tw∈ ⊕w⁰<wZTw⁰.

and we have the product formula, for w, w⁰∈W(1),

(12) Eo(w)Eo•w(w⁰) =qw,w⁰Eo(ww⁰), qw,w⁰ = (qwqw⁰q⁻¹_ww0)^1/2.

The theorem, proved by reduction toq_s= 1, implies that (E_o(w))_w∈W(1) is a basis of HR(q_s, c_s).

Corollary 2.8. TheR-module of basis(E_o(λ))_λ∈Λ(1) is a subalgebra A_o(1)ofH_R(q_s, c_s) with product

Eo(λ)Eo(λ⁰) =qλ,λ⁰Eo(λλ⁰) for λ, λ⁰∈Λ(1).

The Bernstein relations that we are going to present allow us to give a presentation ofHR(qs, cs), starting from the basis (Eo(w))_w∈W₍₁₎ when the (qs)_s∈Saf f/∼ are invertible in R. But the Bernstein relations exist without any conditions on (qs)_s∈Saf f/∼ and their many applications will be developed in the sequel of this article.

Letν : Λ(1)→V be the homomorphism such thatλ∈Λ(1) act onV by translation by ν(λ), and for α ∈Φ let eα be the positive integer such that the set {eαα| α ∈ Φ}

is the reduced root system Σ definingW^{af f}. A rootβ∈Σ taking positive values on the Weyl chamberD⁺of vertexx0containing the alcoveCis called positive. For an arbitrary spherical orientation o of Weyl chamber D_o, let ∆_o be the corresponding basis of the reduced root system Σ (not of Φ), S_o and S be the sets of orthogonal reflections with respect to the walls of D_o and ofD⁺.

Fors∈(S∩S_o)(1) andλ∈Λ(1), the Bernstein relations show that Eo(s)(E_o•s(λ)−Eo(λ)) =Eo(sλs⁻¹)Eo(s)−Eo(s)Eo(λ).

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belongs to Ao(1) and give its expansion on the basis (Eo(λ))_λ∈Λ(1). The proof proceeds by reduction toqs= 1.

For a rootβ in Σ, sβ∈W0 andν(λ) is fixed bysβ if and only if β◦ν(λ) = 0. As the translation byν(λ) stabilizesH, we haveβ◦ν(λ)∈Z. Whenβ◦ν(λ)6= 0, we denote its sign byβ(λ).

Theorem 2.9. (Bernstein relation in the generic algebraHR[q_s)](qs, cs))

Let o be a spherical orientation, s∈(S∩S_o)(1),β ∈∆ such thats∈s_β(1), and let λ∈Λ(1).

Whenβ◦ν(λ) = 0, we have E_o•s(λ) =E_o(λ).

The pro-p-Iwahori Hecke algebra of a reductive p-adic group I