THE AFFINE YOKONUMA–HECKE ALGEBRA AND THE PRO-p-IWAHORI–HECKE ALGEBRA

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THE AFFINE YOKONUMA–HECKE ALGEBRA AND THE PRO-p-IWAHORI–HECKE ALGEBRA

Maria Chlouveraki, Vincent Sécherre

To cite this version:

Maria Chlouveraki, Vincent Sécherre. THE AFFINE YOKONUMA–HECKE ALGEBRA AND THE

PRO-p-IWAHORI–HECKE ALGEBRA. 2015. �hal-01143597�

THE AFFINE YOKONUMA–HECKE ALGEBRA AND THE PRO-p-IWAHORI–HECKE ALGEBRA

MARIA CHLOUVERAKI AND VINCENT S´ECHERRE

Abstract. We prove that the affine Yokonuma–Hecke algebra defined by Chlouveraki and Poulain d’Andecy is a particular case of the pro-p-Iwahori–Hecke algebra defined by Vign´eras.

1. Introduction

A family of complex algebras Y^aff_d,n(q), called affine Yokonuma–Hecke algebras, has been defined and studied by Chlouveraki and Poulain d’Andecy in [ChPo2]. The existence of these algebras has been first mentioned by Juyumaya and Lambropoulou in [JuLa1]. These algebras, which generalise both affine Hecke algebras of type A and Yokonuma–Hecke algebras [Yo], are used to determine the representations of Yokonuma–

Hecke algebras [ChPo1] and construct invariants for framed and classical knots in the solid torus [ChPo2].

Moreover, whenq²is a power of a prime numberpandd=q²−1, one can verify that the affine Yokonuma–

Hecke algebra Y_d,n^aff(q) is isomorphic to the convolution algebra of complex valued and compactly supported functions on the group GLn(F), withF a suitablep-adic field, that are bi-invariant under the pro-p-radical of an Iwahori subgroup (see [Vi1]). It is natural to ask whether there is a family of algebras that generalises, in a similar way, affine Hecke algebras in arbitrary type.

In a recent series of preprints [Vi2, Vi3, Vi4], Vign´eras introduced and studied a large family of algebras, called pro-p-Iwahori–Hecke algebras. They generalise convolution algebras of compactly supported functions on ap-adic connected reductive group that are bi-invariant under the pro-p-radical of an Iwahori subgroup, which play an important role in thep-modular representation theory ofp-adic reductive groups (see [AHHV]

for instance).

In this note, we show that the algebra Y^aff_d,n(q) of Chlouveraki and Poulain d’Andecy is a pro-p-Iwahori–

Hecke algebra in the sense of Vign´eras [Vi2].

2. The affine Yokonuma–Hecke algebra

Letd, n∈Z_>0. Letq be an indeterminate and setR:=C[q, q⁻¹]. We denote byY_d,n^aff(q) the associative algebra overRgenerated by elements

t1, . . . , tn, g1, . . . , gn−1, X1, X₁⁻¹ subject to the following defining relations:

(2.1)

(br1) gigj = gjgi for alli, j= 1, . . . , n−1 such that|i−j|>1, (br2) gigi+1gi = gi+1gigi+1 for alli= 1, . . . , n−2,

(fr1) titj = tjti for alli, j= 1, . . . , n,

(fr2) gitj = tsi(j)gi for alli= 1, . . . , n−1 andj= 1, . . . , n,

(fr3) t^d_j = 1 for allj= 1, . . . , n,

(aff1) X1g1X1g1 = g1X1g1X1

(aff2) X1gi = giX1 for alli= 2, . . . , n−1, (aff3) X1tj = tjX1 for allj= 1, . . . , n.

(quad) g_i² = 1 + (q−q⁻¹)eigi for alli= 1, . . . , n−1,

2010Mathematics Subject Classification. 20C08.

We are grateful to Guillaume Pouchin for his remarks and always helpful explanations. We would also like to thank Lo¨ıc Poulain d’Andecy for our fruitful discussions on the topics of this paper.

1

wheresi denotes the transposition (i, i+ 1) and

(2.2) ei:= 1

d

d−1X

k=0

t^k_it^d−k_i+1

for alli= 1, . . . , n−1. The algebraY_d,n^aff(q) was called in [ChPo1] theaffine Yokonuma–Hecke algebra and was studied in [ChPo2], together with its cyclotomic quotients. This algebra is isomorphic to the modular framisation of the affine Hecke algebra; see definition in [JuLa2, Section 6] and Remark 1 in [ChPo1]. For d= 1,Y_1,n^aff(q) is the standard affine Hecke algebra of typeA.

Remark 2.3. Relations (br1), (br2), (aff1) and (aff2) are the defining relations of the affine braid group B_n^aff. Adding relations (fr1), (fr2) and (aff3) yields the definition of theextended affine braid group orframed affine braid group Zⁿ⋊B_n^aff, with the tj’s being interpreted as the “elementary framings” (framing 1 on thejth strand). The quotient ofZⁿ⋊B_n^aff over the relations (fr3) is themodular framed affine braid group (Z/dZ)ⁿ⋊B_n^aff (the framing of each braid strand is regarded modulod). Thus, the affine Yokonuma–Hecke algebraY_d,n^aff(q) can be obtained as the quotient of the group algebraR[(Z/dZ)ⁿ⋊B_n^aff] over the quadratic relation (quad).

Note that, for alli= 1, . . . , n−1, the elements ei are idempotents, and we havegiei=eigi. Moreover, the elementsgi are invertible, with

(2.4) g⁻¹_i =gi−(q−q⁻¹)ei for alli= 1, . . . , n−1.

Now, fori, l= 1, . . . , n, we set

(2.5) ei,l:= 1

d

d−1X

k=0

t^k_it^d−k_l

The elementsei,l are idempotents. We also haveei,i = 1,ei,i+1=ei andei,l=el,i. Moreover, it is easy to check that

(2.6) tjei,l=tsi,l(j)ei,l=ei,ltsi,l(j)=ei,ltj for allj= 1, . . . , n, wheresi,ldenotes the transposition (i, l).

We define inductively elementsX2, . . . , Xn ofY_d,n^aff(q) by

(2.7) Xi+1:=giXigi fori= 1, . . . , n−1.

We have that the elements t1, . . . , tn, X₁^±1, . . . , X_n^±1 form a commutative family [ChPo1, Proposition 1].

Moreover, in [ChPo1, Lemma 1], it is proved that, for anyi∈ {1, . . . , n}, we have

(2.8) gjXi=Xigj and gjX_i⁻¹=X_i⁻¹gj forj= 1, . . . , n−1 such thatj 6=i−1, i.

Finally, using (2.1)(quad) and (2.4), it is easy to check that, for anyi∈ {1, . . . , n}, we have (2.9) giXi=Xi+1gi−(q−q⁻¹)eiXi+1 and giXi+1=Xigi+ (q−q⁻¹)eiXi+1, which in turn yields

(2.10) giX_i⁻¹=X_i+1⁻¹gi+ (q−q⁻¹)eiX_i⁻¹ and giX_i+1⁻¹ =X_i⁻¹gi−(q−q⁻¹)eiX_i⁻¹. We can easily prove by induction, ona, b∈Z_>0, that the following equalities hold:

(2.11) giX_i^a=X_i+1^a gi−(q−q⁻¹)ei a−1X

k=0

X_i^kX_i+1^a−k and giX_i+1^b =X_i^bgi+ (q−q⁻¹)ei b−1X

k=0

X_i^kX_i+1^b−k

(2.12) giX_i^−a=X_i+1^−agi+ (q−q⁻¹)ei a−1X

k=0

X_i^−a+kX_i+1^−k and giX_i+1^−b =X_i^−bgi−(q−q⁻¹)ei

Xb−1 k=0

X_i^−b+kX_i+1^−k Note also that

(2.13) giXiXi+1 =giXigiXigi=Xi+1Xigi=XiXi+1gi and giX_i⁻¹X_i+1⁻¹ =X_i⁻¹X_i+1⁻¹gi. The above formulas yield it turn the following lemma:

2

Lemma 2.14. We have the following identities satisfied inY_d,n^aff(q)(i= 1, . . . , n−1):

(2.15) giX_i^aX_i+1^b =











X_i^bX_i+1^a gi−(q−q⁻¹)ei a−b−1P

k=0

X_i^b+kX_i+1^a−k ifa≥b,

X_i^bX_i+1^a gi+ (q−q⁻¹)ei b−a−1P

k=0

X_i^a+kX_i+1^b−k ifa≤b,

a, b∈Z.

Let w ∈ S_n, where S_n is the symmetric group on n letters, and let w = si1si2. . . sir be a reduced expression forw. Since the generatorsgiof Y_d,n^aff(q) satisfy the same braid relations, (br1) and (br2), as the generators of S_n, Matsumoto’s lemma implies that the element gw :=gi1gi2. . . gir is well-defined, that is, it does not depend on the choice of the reduced expression forw. We then obtain anR-basis ofY_d,n^aff(q) as follows:

Theorem 2.16. [ChPo2, Theorem 4.15]The set B^aff_d,n=n

t^a₁¹· · ·t^a_nⁿX₁^b1· · ·X_n^bngw |a1, . . . , an∈Z/dZ, b1, . . . , bn∈Z, w∈S_n o

is anR-basis of Y_d,n^aff(q).

Ford= 1,B_1,n^aff is the standard Bernstein basis of the affine Hecke algebra Y_1,n^aff(q) of type A.

3. The pro-p-Iwahori–Hecke algebra

Let Λ denote the abelian group Zⁿ. Forλ= (λ1, . . . , λn), λ^′ = (λ^′₁, . . . , λ^′_n)∈Λ, we have λλ^′ =λ^′λ= (λ1+λ^′₁, . . . , λn+λ^′_n). We denote byλ◦λ^′ the dot product ofλandλ^′, that is, the integerPn

j=1λjλ^′_j. We setεi := (0,0, . . . ,0,1,−1,0, . . . ,0)∈Λ, where 1 is in thei-th position and−1 is in the (i+ 1)-th position, fori= 1, . . . , n−1.

LetW be the extended affine Weyl group Λ⋊S_n of typeA. For all σ∈S_n and λ= (λ1, . . . , λn)∈Λ, we set

σ(λ) :=σλσ⁻¹= (λσ(1), . . . , λσ(n)).

Now, the symmetric groupS_n is generated by the setS={s1, . . . , sn−1}, where si denotes the transpo- sition (i, i+ 1). We set

γ:=sn−1sn−2. . . s2s1∈S_n and h:= (0,0, . . . ,0,1)γ∈W.

Note that we havehsih⁻¹=si−1for alli= 2, . . . , n−1. Sets0:=hs1h⁻¹∈W. Then the setS^aff =S∪{s0} is a generating set of the affine Weyl groupW^aff of typeAand we haveW =hhi⋊W^aff. Moreover, we can extend the length function ℓofW^aff to W by settingℓ(h^kw^aff) =ℓ(w^aff) for allw^aff ∈W^aff,k∈Z.

LetT be a (finite) abelian group such thatS_n acts onT. Like above, for allσ∈S_n andt∈ T, we set σ(t) :=σtσ⁻¹.

We consider the semi-direct product Wf := (T ×Λ)⋊S_n. Every element of fW can be written in the form t λ σ, with t∈ T, λ∈Λ, σ∈S_n. Note thatt and λcommute with each other. We set Λ :=e T ×Λ and Se_n :=T ⋊S_n. We haveWf=ΛeSe_n, withΛe∩Se_n =T. Like above, for allσ∈S_n andν ∈Λ, we sete σ(ν) :=σνσ⁻¹. We also setSe:={ts|s∈S, t∈ T }andSe^aff :={ts|s∈S^aff, t∈ T }. Finally, we can extend the length functionℓofW to fW by settingℓ(tw) =ℓ(w) for allw∈W,t∈ T.

Theorem 3.1. [Vi2, Theorem 2.4]Let R be a ring and let(qs, cs)_s∈Se^aff ∈R×R[T]be such that (a) qs=qts andcts=tcsfor all t∈ T.

(b) qs=qs^′ andcs^′ =wcsw⁻¹ if s^′=wsw⁻¹ for somew∈Wf.

Then the freeR-moduleHR(qs, cs)of basis(Tw)_w∈fW has a uniqueR-algebra structure satisfying:

• The braid relations: TwTw^′ =Tww^′ ifw, w^′∈Wf,ℓ(w) +ℓ(w^′) =ℓ(ww^′).

• The quadratic relations: T_s²=qss²+csTsfor s∈Se^aff.

3

The algebraHR(qs, cs) is called thepro-p-Iwahori–Hecke algebra of fW overR. If (qs)_s∈Se^aff are taken to be indeterminates satisfying (a) and (b) above,R:=R[qs^1/2, q^−1/2s ] and (cs)_s∈Se^aff ∈R[T] satisfy (a) and (b) above, thenH^R(qs, cs) is called thegeneric pro-p-Iwahori–Hecke algebra of fW. Note that, in this case, since qsis invertible for alls∈Se^aff,Tsis invertible inH^R(qs, cs), with

(3.2) T_s⁻¹=q_s⁻¹s⁻²(Ts−cs).

We deduce that every elementTw, forw∈Wf, is invertible inHR(qs, cs). Note that any pro-p-Iwahori–Hecke algebra ofWfcan be obtained from the generic pro-p-Iwahori–Hecke algebraH^R(qs, cs) with a specialisation of parameters. Further, by replacing the generatorsTs by ¯Ts:=qs^−1/2Ts, the quadratic relations become

T¯_s²=s²+q^−1/2s csT¯s for all s∈Se^aff.

Thus, if (qs^1/2)_s∈Se^aff are chosen so that they also satisfy conditions (a) and (b) of Theorem 3.1, we obtain an isomorphism betweenH^R(qs, cs) andH^R(1, q^−1/2s cs). Therefore, from now on, without loss of generality, we will assume thatqs= 1 for alls∈Se^aff. ThenR=R. We now have the following Bernstein presentation of HR(1, cs).

Theorem 3.3. [Vi2, Theorem 2.10] The R-algebra HR(1, cs) is isomorphic to the free R-module of basis (E(w))_w∈fW endowed with the uniqueR-algebra structure satisfying:

• Braid relations: E(w)E(w^′) =E(ww^′)for w, w^′ ∈Se_n,ℓ(w) +ℓ(w^′) =ℓ(ww^′).

• Quadratic relations: E(s)²=s²+csE(s) fors∈S.e

• Product relations: E(ν)E(w) =E(νw)for ν∈Λ,e w∈Wf.

• Bernstein relations: Fors=tsi∈Se(t∈ T,i= 1, . . . , n−1) and ν=τ λ∈Λe (τ∈ T,λ∈Λ), E(s(ν))E(s)−E(s)E(ν) =

( 0 if si(λ) =λ,

ǫi(λ)csP|εi◦λ|−1

k=0 E(τ µi(k, λ)) if si(λ)6=λ, whereǫi(λ) = sign(εi◦λ)∈ {1,−1}and

µi(k, λ) =

λ ε^k_i ifǫi(λ) =−1 si(λ)ε^k_i ifǫi(λ) = 1.

Remark 3.4. Note that, in the above presentation, we takeE(s) =Tsfor alls∈Se(in particular,E(t) =t for allt∈ T).

4. Main result

Let q be an indeterminate. Our aim in this section will be to show that the affine Yokonuma–Hecke algebraY_d,n^aff(q) is isomorphic to the pro-p-Iwahori–Hecke algebraHR(qs, cs) ofWfwhen we take

• T = (Z/dZ)ⁿ =ht1, . . . , tn |t^d_j = 1, titj =tjti, for alli, j= 1, . . . , ni;

• R=R=C[q, q⁻¹] ;

• qtsi= 1 for alli= 0,1, . . . , n−1,t∈(Z/dZ)ⁿ ;

• ctsi= (q−q⁻¹)t ei for alli= 0,1, . . . , n−1,t∈(Z/dZ)ⁿ, where e0:=e1,n.

We then havefW = (Z/dZ×Z)ⁿ⋊S_n={t λ σ|t∈(Z/dZ)ⁿ, λ∈Λ, σ∈S_n}. The action ofS_n on (Z/dZ)ⁿ is given by

σ(tj) =σtjσ⁻¹=t_σ(j) for allσ∈S_n, j= 1, . . . , n.

The action ofS_n extends linearly to the group algebraR[(Z/dZ)ⁿ].

First we check that the assumptions (a) and (b) of Theorem 3.1 are satisfied in this case. Let s∈Se^aff = {tsi|i= 0,1, . . . , n−1, t∈(Z/dZ)ⁿ}. By definition, we haveqs=qts= 1 andcts=tcsfor allt∈(Z/dZ)ⁿ.

Now, set

λ0:= (−1,0, . . . ,0,1)∈Λ and λi:= (0,0, . . . ,0,0)∈Λ for alli= 1, . . . , n−1.

4

Moreover, letσ0 denote the transposition (1, n) =s1,nandσi denote the transposition (i, i+ 1) =si for all i= 1, . . . , n−1. We then have

si=λiσi for alli= 0,1, . . . , n−1.

So

Se^aff ={tλiσi|i= 0,1, . . . , n−1, t∈(Z/dZ)ⁿ}.

Lets, s^′∈Se^aff be conjugate infW. By definition, we haveqs=qs^′ = 1. Now, let us writes=tλiσi and s^′ =t^′λi^′σi^′ fori, i^′∈ {0,1, . . . , n−1} andt, t^′ ∈(Z/dZ)ⁿ. Moreover, let w∈Wf be such thats^′=wsw⁻¹, and writew=τ λσ withτ ∈(Z/dZ)ⁿ,λ∈Λ andσ∈S_n. Then

t^′λi^′σi^′ = (τ λσ)(tλiσi)(σ⁻¹λ⁻¹τ⁻¹) = [τ σ(t)(σσiσ⁻¹)(τ⁻¹)][λσ(λi)(σσiσ⁻¹)(λ⁻¹)][σσiσ⁻¹].

We deduce that

σi^′ =σσiσ⁻¹ and t^′=τ σ(t)(σσiσ⁻¹)(τ⁻¹) =τ σ(t)σi^′(τ⁻¹).

By (2.6), we have

(4.1) t^′ei^′ =τ σ(t)σi^′(τ⁻¹)ei^′ =τ σ(t)τ⁻¹ei^′ =σ(t)ei^′ =σ(t)ww⁻¹ei^′ =wtw⁻¹ei^′. Furthermore, we have

σi^′σ(i) =σσi(i) =σ(i+ 1) and σi^′σ(i+ 1) =σσi(i+ 1) =σ(i),

where iand i+ 1 in the above equalities are considered modulo nif necessary. If σ(i) is fixed byσi^′, then the first equality implies thatσ(i) =σ(i+ 1), which is absurd. Similarly, ifσ(i+ 1) is fixed byσi^′, then the second equality implies thatσ(i+ 1) =σ(i), which is absurd. We deduce that

{σ(i), σ(i+ 1)}=

{i^′, i^′+ 1} if 1≤i^′≤n−1 ; {1, n} ifi^′ = 0 ;

which in turn yields

(4.2) weiw⁻¹=σ(ei)ww⁻¹=σ(ei) =ei^′. Combining (4.1) and (4.2), we obtain

cs^′ = (q−q⁻¹)t^′ei^′ = (q−q⁻¹)wtw⁻¹weiw⁻¹=w (q−q⁻¹)tei

w⁻¹=wcsw⁻¹.

Therefore, we can define the pro-p-Iwahori–Hecke algebraHR(qs, cs) offW, which is the freeR-module of basis (E(w))_w∈fW endowed with the uniqueR-algebra structure satisfying the relations described explicitly in the previous section.

Theorem 4.3. The R-linear map ϕ:Y_d,n^aff(q)→ HR(qs, cs)defined by

ϕ(t^a₁¹· · ·t^a_nⁿX₁^b1· · ·X_n^bngw) =E(t^a₁¹· · ·t^a_nⁿ(b1, . . . , bn)w), for alla1, . . . , an∈Z/dZ,b1, . . . , bn∈Z,andw∈S_n, is anR-algebra isomorphism.

Proof. The map ϕ is obviously a bijectiveR-linear map, so we just need to show that ϕis an R-algebra homomorphism. For this, it is enough to check that the defining relations (2.1) ofY_d,n^aff(q) are satisfied by the images of its generators viaϕ, that is, the elements

ϕ(t1), . . . , ϕ(tn), ϕ(g1), . . . , ϕ(gn−1), ϕ(X1), ϕ(X₁⁻¹).

First of all, note that ϕ(X₁⁻¹) = E((−1,0, . . . ,0)) = E((1,0, . . . ,0))⁻¹ = ϕ(X1)⁻¹, due to the product relations. The product relations also imply (aff3). Moreover, due to the braid relations, we have immediately that (br1), (br2), (fr1) and (fr3) are satisfied. We also obtain

ϕ(gi)ϕ(tj) =E(si)E(tj) =E(sitj) =E(ts_i(j)si) =E(ts_i(j))E(si) =ϕ(ts_i(j))ϕ(gi) for alli= 1, . . . , n−1 andj= 1, . . . , n, so (fr2) holds.

Now, fori= 1, . . . , n−1, we have

ϕ(gi)²=E(si)²=s²_i +csiE(si) = 1 + (q−q⁻¹)eiE(si) = 1 + (q−q⁻¹)eiϕ(gi),

so (quad) is satisfied byϕ(gi). We will use the Bernstein relations for the remaining defining relations, (aff1) and (aff2).

5

First, note that (1,0, . . . ,0) is fixed by the action ofsifor alli= 2, . . . , n−1. We thus obtain ϕ(X1)ϕ(gi) =E((1,0, . . . ,0))E(si) =E(si(1,0, . . . ,0)s⁻¹_i )E(si) =E(si)E((1,0, . . . ,0)) =ϕ(gi)ϕ(X1).

This yields (aff2).

Now, (1,0, . . . ,0) is not fixed by the action ofs1. By the Bernstein relations, we have E((0,1, . . . ,0))E(s1)−E(s1)E((1,0, . . . ,0)) =cs1E((0,1, . . . ,0)), and thus,

E(s1)E((1,0, . . . ,0)) =E((0,1, . . . ,0))E(s1)−cs1E((0,1, . . . ,0)) =E((0,1, . . . ,0))(E(s1)−cs1).

By (3.2) and Remark 3.4, we have thatE(s1)−cs1 =E(s1)⁻¹, and so E(s1)E((1,0, . . . ,0)) =E((0,1, . . . ,0))E(s1)⁻¹. We obtain

ϕ(X1)ϕ(g1)ϕ(X1)ϕ(g1) = E((1,0, . . . ,0))E(s1)E((1,0, . . . ,0))E(s1)

= E((1,0, . . . ,0))E((0,1, . . . ,0))E(s1)⁻¹E(s1)

= E((1,0, . . . ,0))E((0,1, . . . ,0)).

= E((1,1, . . . ,0)).

and ϕ(g1)ϕ(X1)ϕ(g1)ϕ(X1) = E(s1)E((1,0, . . . ,0))E(s1)E((1,0, . . . ,0))

= E((0,1, . . . ,0))E(s1)⁻¹E(s1)E((1,0, . . . ,0))

= E((0,1, . . . ,0))E((1,0, . . . ,0)).

= E((1,1, . . . ,0)).

Thus, (aff1) also holds, andϕis a bijectiveR-algebra homomorphism, as required.

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[Vi4] M.-F. Vign´eras,The pro-p-Iwahori Hecke algebra of a reductivep-adic group III, preprint (2014).

[Yo] T. Yokonuma,Sur la structure des anneaux de Hecke d’un groupe de Chevalley fini, C. R. Acad. Sci. Paris Ser. I Math.264 (1967) 344–347.

Laboratoire de Math´ematiques de Versailles, Universit´e de Versailles St-Quentin, Bˆatiment Fermat, 45 avenue des ´Etats-Unis, 78035 Versailles cedex, France.

E-mail address:maria.chlouveraki@uvsq.fr

Laboratoire de Math´ematiques de Versailles, Universit´e de Versailles St-Quentin, Bˆatiment Fermat, 45 avenue des ´Etats-Unis, 78035 Versailles cedex, France.

E-mail address:vincent.secherre@uvsq.fr

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