On a conjecture about cellular characters for the complex reflection group $G(d,1,n)$

BLAISE PASCAL

Abel Lacabanne

On a conjecture about cellular characters for the complex reflection group

Volume 27, n^o1 (2020), p. 37-64.

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Publication éditée par le laboratoire de mathématiques Blaise Pascal de l’université Clermont Auvergne, UMR 6620 du CNRS

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On a conjecture about cellular characters for the complex reflection group G(d, 1, n)

Abel Lacabanne

Abstract

We propose a conjecture relating two different sets of characters for the complex reflection group G(d,1,n). From one side, the characters are afforded by Calogero–Moser cells, a conjectural generalisation of Kazhdan–Lusztig cells for a complex reflection group. From the other side, the characters arise from a leveldirreducible integrable representations ofU_q(sl_∞). We prove this conjecture in some cases: in full generality forG(d,1,2)and for generic parameters forG(d,1,n).

Using Cherednik algebras and Calogero–Moser spaces, Bonnafé and Rouquier devel- oped in [4] the notions of cells (right, left or two-sided) and of cellular characters of a complex reflection groupW. Whenever this group is a Coxeter group, they conjectured in [4, Chapter 15] that these notions coincide with the corresponding notions in the Kazhdan–Lusztig theory. As for Hecke algebras with unequal parameters, these notions heavily depend on some parametercdefined on the reflections ofW and invariant by conjugation. In this paper, we are mainly interested in the notion of cellular characters for the complex reflection groupG(d,1,n). Ifd=1, the groupG(d,1,n)is nothing else than the Weyl group of typeAn−1, and ifd =2, we recover the Weyl group of typeBn. Bonnafé and Rouquier showed that if the Calogero–Moser space with parametercassociated to W is smooth then the cellular characters are irreducible. This implies that their notion of Calogero–Moser cellular characters coincides with the notion of Kazhdan–Lusztig cellular characters in typeA. Even in typeB, we only have a complete description for B2[4, Chapter 19]. For the dihedral groupG(d,d,2), a description of Calogero–Moser families and of Calogero–Moser cellular characters has been given by Bonnafé [3], and these are compatible with Kazhdan–Lusztig theory.

Lusztig defined in [13, Chapter 22] a notion of constructible characters of a Coxeter group, using the so-called truncated induction. He conjectured that these constructible characters are exactly the characters carried by the Kazhdan–Lusztig left cells, and proved the result in the equal parameter case. These characters surprisingly appeared in the work of Leclerc and Miyachi [10]. They obtained a closed formula for canonical bases of a level 2 irreducible integrable representationV(Λ_r1+Λ_r2)ofU_q(sl∞)(here theΛ_iare the fundamental weights). By evaluating these expressions atq =1, Leclerc and Miyachi

The author was supported by the Fonds de la Recherche Scientifique - FNRS under Grant no. MIS-F.4536.19.

Keywords:Cellular characters, Complex reflection groups.

2020Mathematics Subject Classification:20F55, 20G42.

retrieved Lusztig’s constructible characters for Weyl groups of typeBandD. With a level dirreducible integral representationV Íd

i=1Λ_ri

, one obtains some characters of the complex reflection groupG(d,1,n)in a similar manner and Leclerc and Miyachi asked whether these characters are a good analogue of constructible characters in typeB.

Thus, we have two sets of characters for the complex reflection groupG(d,1,n), namely the Calogero–Moser cellular characters and the constructible characters of Leclerc and Miyachi. Both sets of characters depend heavily on some parameters (corr=(r₁, . . . ,r_d)), and up to a suitable change of parameters, we conjecture that these two sets of characters are equal.

Conjecture A (Conjecture 4.1). Letrbe a d-tuple of integers. Then there exists an explicit choice of parametercfor the complex reflection groupG(d,1,n)such that the set of Calogero–Moserc-cellular characters and the set of Leclerc–Miyachir-constructible characters coincide.

We refer to the statement in Section 4 for the precise relation between the parametersc andr. The main result of this paper is a proof of this conjecture in two different cases.

Theorem A(Theorem 4.2). Conjecture A is true in the following two cases:

(1) forG(d,1,2)and any parameters, (2) forG(d,1,n)and generic parameters.

We refer to Corollary 1.11 for the notion of generic parameters. To support this conjecture, it would be interesting to retrieve some known properties of the Calogero–

Moser cellular characters, for example the fact that for any Calogero–Moser cellular character there exists a unique irreducible constituent with minimalb-invariant, and its multiplicity is one. This fact is already known for constructible characters of an irreducible finite Coxeter group [2].

The paper is organized as follows. In the first Section, we define the first set of characters we are interested in, the Calogero–Moser cellular characters. There are several equivalent definitions of these characters in [4] and we choose to use a definition using the so-called Gaudin algebra, which is a commutative subalgebra of the group algebra of G(d,1,n)over a localization of a polynomial ring. Using this definition, we give another proof of the irreducibility of Calogero–Moser cellular characters for a parameter outside of the essential hyperplanes defined by Chlouveraki. We rely on some results proven in the Appendix. In Section 2, we set up notation and define the second set of characters we are interested in, the Leclerc–Miyachi constructible characters. We show that in the asymptotic situation, these characters are irreducible. In the third section, we compute

explicitly the Calogero–Moser cellular and Leclerc–Miyachi constructible characters for the complex reflection groupG(d,1,2). On the Calogero–Moser side, we diagonalize the action of the Gaudin algebra on representations ofG(d,1,2)and on the Leclerc–Miyachi side, we compute the canonical bases using the algorithm introduced by Leclerc and Toffin in [11]. Finally, in the last section, we state precisely the conjecture relating these characters, and show that it is valid forG(d,1,2)and any choice of parameters, and for G(d,1,n)for generic parameters.

1.

Calogero–Moser cellular characters

We introduce the set of Calogero–Moserc-cellular characters of a complex reflection group, which we define using the notion of Gaudin algebra, see [4]. In the specific case of G(d,1,n), we introduce a commutative subalgebraJM_cgenerated by the so-called Jucys–

Murphy elements. Using results of the Appendix, we show that the cellular characters of G(d,1,n)for the algebraJM_care sums of Calogero–Moser characters. For specific values ofc, we show that the cellular characters ofG(d,1,n)for the algebraJM_care irreducible, then so are the Calogero–Moserc-cellular characters.

1.1.

Notations

We fixVa finite dimensionalC-vector space, denote by det : GL(V) →C^∗the determinant and byh·,·i:V×V^∗→Cthe duality betweenV and the spaceV^∗of linear forms onV.

We choose for each positive integerdad-th root of unityζ_dsuch thatζ_d^d/l =ζ_lfor alll dividingd. The group ofd-th roots of unity will be denoted byµ_d.

LetW ⊂GL(V)be a finite complex reflection group. We denote by Ref(W)the set of pseudo-reflections ofWand for eachs∈Ref(W), we chooseα_s∈V^∗andα^∨_s ∈Vsuch that

ker(s−IdV)=ker(α_s) and Im(s−IdV)=Cα^∨_s.

We denote byAthe set of reflecting hyperplanes ofW as well as byV^regthe open subsetV\Ð

H∈ AH. A theorem of Steinberg [6, Theorem 4.7] shows thatV^regis the subset of elements ofV with trivial stabilizers with respect to the action ofW.

ForH ∈ A, the pointwise stabilizerW_H ofHis a cyclic group of ordere_H with a chosen generators_H ∈Ref(W). IfΩ∈ A/W, we denote byeΩthe common value ofe_H forH ∈Ω. With these notations, the set of reflections ofWis

Ref(W)=n s^j_H

H ∈ A,1 ≤j ≤e_H−1o , and two reflectionss_H^j ands^j

0

H⁰are conjugate if and only if the hyperplanesHandH⁰are in the same orbit under the action ofW andj=j⁰.

We also fixc: Ref(W) →C,s7→c_sa function which is invariant by conjugation. For anyH∈ Aand 0≤i ≤eH−1, we define

k_H,i = 1 e_H

eH−1

Õ

k=1

ζ_e^k(1−i)_H c_sk H, which satisfy ÍeH−1

i=0 k_H,i = 0; we will often consider indices modulo eH and set k_Ω,i =k_H,iforΩ∈ A/W and anyH∈Ω. We recover the functioncvia

c_sⁱ

H =

eH−1

Õ

j=0

ζ_e^i(j−1)_H k_H,j.

1.2.

Gaudin algebra and Calogero–Moser cellular characters

For any y ∈ V, we define an elementD_y in the group ring ofW with coefficients in C[V^reg]:

D_y= Õ

s∈Ref(W)

c_sdet(s)hy, α_si α_s s.

The Gaudin algebra Gauc(W)is the sub-C[V]-algebra ofC[V^reg]Wgenerated by(D_y)_y∈V. Proposition 1.1([4, 13.4.B]). The algebraGauc(W)is commutative.

Therefore we are in the situation of the Appendix if we setE =CW,A=CWacting on Eby left multiplication,P=C[V^reg]andDi =D_yiacting onEby right multiplication, where(y_i)_i is a basis ofV. The following is Definition A.1 in our setting.

Definition 1.2. The set of cellular characters for the algebra Gauc(W)is called the set of Calogero–Moserc-cellular characters, or for shortc-cellular characters. Thec-cellular character associated toL∈Irr(C(V)Gauc(W))is

γ_L^Gauc(W)= Õ

χ∈Irr(W)

h

Res^C(V)W

C(V)Gauc(W)(C(V)V_χ): L iχ,

whereV_χis a representation ofWaffording the character χand[X: L]is the multiplicity ofLin the moduleX.

Remark 1.3. IfWis a Coxeter group andchas positive values, there is a notion ofc-cellular characters arising from the Kazhdan–Lusztig theory of Hecke algebras. It is conjectured by Bonnafé and Rouquier [4, 15.2, Conjecture L] that the set of Kazhdan–Lusztig cellular characters and of Calogero–Moser characters coincide.

For all z ∈ V^reg, we can specialize D_y to an element of CW, by evaluating the coefficients inC[V^reg]atz:

Õ

s∈Ref(W)

csdet(s)hα_s,yi

hα_s,zis∈CW.

Choosingz=y, one obtains a central element ofCW, called the Euler element, which does not depend ony:

eu_c= Õ

s∈Ref(W)

c_sdet(s)s.

Lemma 1.4. Let χ ∈Irr(W). GivenΩ ∈ A/W andH ∈ Ω, we definem_{Ω, χ}^j =hχ|WH, det^−j_|W

Hi_WH, whereh ·,· i_WH denotes the scalar product of characters ofW_H. The Euler elementeu_cacts on a representation affording the character χby multiplication by

Õ

Ω∈ A/W eΩ

Õ

j=0

|Ω|eΩm_{Ω, χ}^j χ(1) kΩ,j.

Proof. See [4, Lemma 7.2.1].

1.3.

The imprimitive reflection group

In this subsection,Vis of dimensionnwith a chosen basis(y₁, . . . ,y_n)with dual basis (x₁, . . . ,xn). Using this choice of basis, we identify GL(V)to GLn(C). We also fix a positive integerdand denote byζthed-th root of unityζ_d.

1.3.1. The groupG(d,1,n)and its reflections

It is easy to describe the groupG(d,1,n)in terms of matrices: it is the subgroup of GLn(C) with elements the monomial matrices with coefficients inµ_d. The permutation matrix corresponding to the transposition(i j)will be denoted bys_i,j and the diagonal matrix with diagonal entries(1, . . . ,1, ζ,1, . . . ,1), withζ at thei-th position will be denoted byσ_i.

The set of reflections ofG(d,1,n)splits intodconjugacy classes:

Ref(G(d,1,n))=

d−1

Ä

k=0

Ref(G(d,1,n))_k, where Ref(G(d,1,n))₀ = σ_i^rs_i,jσ_i^−r

1≤i< j ≤n,0≤r≤d−1 and for 1 ≤ k ≤ d−1, Ref(G(d,1,n))_k =σ_i^k

1≤i≤n .

We now give an explicit choice forα_sandα_s^∨for any reflections. For the reflection s_i,j,r =σ_i^rs_i,jσ_i^−r, the reflecting hyperplaneH_i,j,r is given by the kernel of the linear

formα_i,j,r =x_i −ζ^rx_j and the eigenspace associated to the eigenvalue−1 is spanned byα^∨_i,j,r =ζ^ry_i−y_j. For the reflectionσ_i^k, the reflecting hyperplaneHiis given by the kernel of the linear formα_i =x_iand the eigenspace associated to the eigenvalueζ^k is spanned byα_i^∨=y_i.

Under the action ofG(d,1,n), the set of reflecting hyperplanesAhas only two orbits, Ω₀ =

H_i,j,r

1≤i <j ≤n,0≤r <d−1 andΩ₁ = {H_i |1≤i ≤n} which are of respective cardinaldⁿ⁽ⁿ₂⁻¹⁾andn.

Given a functionc: Ref(W) →Cconstant on the conjugacy classes, we denote its value on Ref(G(d,1,n))_k byc_k and will writek_i instead of kΩ1,i. We will try not to introduce the parameters kΩ0,0 and kΩ0,1 which are respectively equal to−^c₂⁰ and ^c₂⁰. Finally, the Euler element associated toG(d,1,n)andcwill be denoted byeu_c,n. 1.3.2. Representations andd-partitions

The representation theory ofG(d,1,n)is well known and is governed by the combinatorics ofd-partitions ofn, see [8, Section 5.1] for example. A partition ofnis a finite sequence of integers λ = (λ1, . . . , λ_r) adding up to n such that λ1 ≥ λ2 ≥ · · ·λ_r > 0, and we set|λ| = n. A d-partition ofn is a d-tuple(λ⁽¹⁾, . . . , λ^(d)) of partitions such that Íd

i=1|λ⁽ⁱ⁾| = n. The isomorphism classes of irreducible complex representations of G(d,1,n)are parameterized byd-partitions ofn, and for such ad-partitionλ, we denote byV_λthe corresponding representation.

One can describe the branching ruleG(d,1,n) ⊂ G(d,1,n+1)in terms of Young diagrams. The Young diagram[λ]of ad-partitionλofnis the set

n(a,b,c) ∈Z>0×Z>0× {1, . . . ,d}

1≤b≤λ^(c)_a o ,

whose elements will be called boxes. The content cont(γ)of a boxγ =(a,b,c)is the integerb−a. A boxγof[λ]is said to be removable if[λ] \ {γ}is the Young diagram of ad-partitionµofn−1, and in this case, the boxγis said to be addable toµ.

Proposition 1.5([8, Proposition 5.1.8]). Letλbe ad-partition ofn. Then Ind^G(d,1,n+1)_G(d,1,n) (V_λ)=Ê

µ

V_µ

where µruns over thed-partitions ofn+1with Young diagram obtained by adding an addable box to the Young diagram ofλ. Concerning the restriction,

Res^G(d,1,n)_G(d,1,n−1)(V_λ)=Ê

µ

V_µ

whereµruns over thed-partitions ofn−1with Young diagram obtained by removing a removable box from the Young diagram ofλ.

Using this branching rule, we define a basis ofV_λin terms of standardλ-tableaux, which are bijectionst:[λ] → {1, . . . ,n}such that for all boxesγ=(a,b,c)andγ⁰=(a⁰,b⁰,c) we havet(γ) < t(γ⁰)if a =a⁰andb < b⁰ora < a⁰and b = b⁰. Giving a standard λ-tableau is then equivalent to giving a sequence ofd-partitions(λ^t[i])_1≤i≤nsuch that [λ^t[i]] = t⁻¹({1, . . . ,i}). Therefore V_λ is the direct sum of one dimensional spaces Dt, where for all 1 ≤ i ≤ n the space Dt is in the irreducible componentV_λt[i] of Res^G_G(d,1,i)^(d,1,n)(V_λ).

1.3.3. A commutative subalgebra ofCG(d,1,n)

For 1≤k ≤n, we define the following elements ofCG(d,1,n): J_k =eu_c,k−eu_c,k−1= Õ

s∈Ref(G(d,1,k)) s<Ref(G(d,1,k−1))

c_sdet(s)s.

Ifd=1, these elements are the usual Jucys–Murphy elements for the symmetric group S_k, multiplied by the scalarc₀.

Lemma 1.6. For all1≤i,j ≤n, the Jucys–Murphy elementsJ_i andJ_j commute and J_i+1 =s_i,i+1,0J_is_i,i+1,0−c₀Íd−1

r=0 s_i,i+1,r.

Proof. It is immediate to check that the Jucys–Murphy elementJ_iis equal to J_i =−c₀ Õ

1≤p<i d−1

Õ

r=0

s_p,i,r+

d−1

Õ

r=1

c_rζ^rσ_i^r.

Since the conjugate of s_p,i,r bys_i,i+1,0iss_p,i+1,r and the conjugate ofσ_i^r bys_i,i+1,0is σ_i+1^r , we have that

s_i,i+1,0J_is_i,i+1,0⁻¹ =−c₀ Õ

1≤p<i d−1

Õ

r=0

s_p,i+1,r+

d−1

Õ

r=1

c_rζ^rσ_i+1^r ,

and we obtain the induction formula.

Sinceeu_c,icommutes with every element ofCG(d,1,i), one see thatJ_i =eu_c,i−eu_c,i−1 commutes with every element ofCG(d,1,i−1), and therefore withJ₁, . . . ,J_i−1. The commutative subalgebra ofCG(d,1,n)generated by the elementsJ1, . . . ,Jnis denoted byJM_c(d,n). On the representationV_λ, the action of the Jucys–Murphy elements is simultaneously diagonalizable, and one can easily compute the eigenvalues using Lemma 1.4.

Proposition 1.7. Letλbe ad-partition ofnandγ =(a,b,c)a removable box of[λ].

ThenJ_nacts on the componentV_λ\{γ}ofRes^G(d,1,n)_G(d,1,n−1)(V_λ)by multiplication by the scalar d(k_1−c−c₀(b−a)).

Proof. We start by computing the action of the Euler element onVλ, and therefore we need the values of the integersm_Ω,j^χλ, forΩ∈ A/Wand 0≤ j ≤e_eΩ−1, whereχ_λis the character ofV_λ. From [14, Lemma 6.1] we have

1

χ_λ(1)h(χ_λ)|_hs

0i,det^jihs0i= |λ^(j+1)| n

and 1

χ_λ(1)h(χ_λ)|_hs

1i,detihs1i= 1

2− 1

n(n−1) Õ

γ∈[λ]

cont(γ).

Therefore, using Lemma 1.4,eu_c,nacts onV_λby multiplication by the scalar ω_λ(eu_c,n)=

d−1Õ

j=0

k−j

dn

χ_λ(1)h(χ_λ)|_hs

0i,det^jihs0i

+c0

2

dn(n−1)

χ_λ(1) (h(χ_λ)_|hs

1i,deti_hs1i− h(χ_λ)_|hs

1i,1i_hs1i).

Buth(χ_λ)|_hs

1i,1ihs1i= χ_λ(1) − h(χ_λ)|_hs

1i,detihs1i, so that ω_λ(eu_c,n)=d

d−1

Õ

j=0

k_−j|λ^(j+1)| −dc₀ Õ

γ∈[λ]

cont(γ)

and we obtain the desired formula becauseJ_n=eu_c,n−eu_c,n−1. Corollary 1.8. Letλbe ad-partition ofn,tbe a standardλ-tableau and1≤k ≤n. The element Jkacts onDtby multiplication byd(k_1−c−c0(b−a)), wheret⁻¹(k)=(a,b,c).

1.3.4. Cellular characters for JM_c(d,n)and c-cellular characters

As well as for the Gaudin algebra, we define cellular characters for the algebraJM_c(d,n). We are again in the situation of the Appendix if we setE=CG(d,1,n),A=CG(d,1,n) acting onEby left multiplication,P=CandD_i =J_i acting onEby right multiplication.

The following is Definition A.1 in our setting.

Definition 1.9. Thec-cellular character associated toL∈Irr(JMc(d,n))is γ^JM_L ^c(d,n)= Õ

χ∈Irr(W)

h

Res^CG_JM^(d,1,n)

c(d,n) (V_χ): L i χ,

whereV_χis a representation ofWaffording the character χ.

These characters are easier to compute than thec-cellular characters, sinceJM_c(d,n) is commutative and split, and are close toc-cellular characters in the following sense.

Theorem 1.10. Every cellular character for the algebraJM_c(d,n)is a sum of Calogero–

Moserc-cellular characters.

Proof. We denote by Gau⁽⁰⁾_c (d,n)the sub-C[V]-algebra ofC[V^reg]G(d,1,n)generated by x_kD_yk =

d−1

Õ

r=1

c_rζ^rσ_k^t−c₀

d−1

Õ

r=0

©

­

« Õ

1≤i<k

−ζ^rx_k

x_i−ζ^rx_ks_i,k,r+ Õ

k<j≤n

x_k

x_k−ζ^rx_js_k,j,rª

®

¬ , for 1≤k≤n. Sincex_k is invertible inC(V)for allk, the algebrasC(V)Gauc(G(d,1,n)) andC(V)Gau⁽⁰⁾_c (d,n)are equal, and thec-cellular characters are therefore equal to the cellular characters for Gau⁽⁰⁾_c (d,n). We then specialize the algebra Gau⁽⁰⁾_c (d,n)with respect to the following increasing sequence of prime ideals

0⊂p₁ ⊂p₂ ⊂ · · · ⊂p_n

wherep_i is the prime ideal ofC[V]generated byx₁, . . . ,x_i. We define recursively the algebra Gau⁽ⁱ⁾_c (d,n)by

Gau⁽ⁱ⁾_c (d,n)=Gau⁽ⁱ⁻¹⁾_c (d,n)/x_iGau⁽ⁱ⁻¹⁾_c (d,n),

and we denote byπ⁽ⁱ⁾: Gau⁽ⁱ⁾_c (d,n) → Gau⁽ⁱ⁺¹⁾_c (d,n)the corresponding quotient map, andΠ⁽ⁱ⁾=π⁽ⁱ⁻¹⁾◦ · · · ◦π⁽⁰⁾. The algebra Gau⁽ⁿ⁾_c (d,n)is hence a subalgebra ofCG(d,1,n). By Proposition A.3, the cellular characters for the algebra Gau⁽ⁱ⁾_c (d,n)are sums of cellular characters for the algebra Gau⁽ⁱ⁻¹⁾_c (d,n), and therefore the cellular characters for the algebra Gau⁽ⁱ⁾_c (d,n)are sums ofc-cellular characters.

An easy induction shows that ifi≥kthen Π⁽ⁱ⁾(x_kD_k)=

Õd−1

r=1

c_rζ^rσ_k^r−c₀ Õd−1

r=0

Õ

1≤j<k

s_j,k,r,

and ifi <kthen Π⁽ⁱ⁾(x_kD_k)=

Õd−1

r=1

crζ^rσ_k^r

−c0 d−1Õ

r=0











 Õi

j=1

s_j,k,r+ Õ

i<j<k

−ζ^rxk

x_j−ζ^rx_ks_j,k,r+ Õ

k<j≤n

xk

x_k−ζ^rx_js_k,j,r











 . The algebra Gau⁽_cⁿ⁾(d,n)is thus equal toJM_c(d,n)sinceΠ⁽ⁿ⁾(x_kD_k)= Jk, which

ends the proof.

As a Corollary, we retrieve the fact that the c-cellular characters are generically irreducible, see [4, Theorem 14.4.1] or [1, Theorem 10 (3)].

Corollary 1.11. Suppose that the parametercis such that c0,0 and (k_p−kq) −c0j ,0,

for all1≤p,q≤dand−n<j <n. Then the Calogero–Moserc-cellular characters ofG(d,1,n)are irreducible.

Proof. SinceJM_c(d,n) ⊂CG(d,1,n), any simple representation occurs in someV_λand is therefore of the formDt forta standardλ-tableau. It suffices to show that these one dimensional representations ofJM_c(n,d)are pairwise non-isomorphic. Indeed, ifλand

µare differentd-partitions we have Res^G(d,1,n)_JM

c(d,n)(V_λ)=Ê

t

Dt and Res^G(d,1,n)_JM

c(d,n)(V_µ)=Ê

s

Ds,

wheret(resp.s) runs in the set of standardλ-tableaux (resp. µ-tableaux). Therefore the restrictions ofV^λandV^µtoJM_c(d,n)have no common constituent ifλ,µand for any standardλ-tableau, the multiplicity ofDtin Res^G_JM^(d,1,n)

c(d,n)(V_λ)is 1.

Lettbe a standardλ-tableau andt⁰be a standardλ⁰-tableau, whereλandλ⁰are two non necessarily differentd-partitions ofn. We suppose thatt,t⁰and we prove that the sequences

(k₁−cp −c₀(b_p−a_p))₁≤p≤n and (k₁−c⁰p −c₀(b⁰_p−a⁰_p))₁≤p≤n

are different, where(a_p,bp,cp)=t⁻¹(p)and(a⁰_p,b⁰_p,c⁰_p)=(t⁰)⁻¹(p). Let 1≤p≤nbe the minimal integer such thatt⁻¹(p)and(t⁰)⁻¹(p)are different. Denote byµthe common partitionλ^t[p−1]=(λ⁰)^t0[p−1].

Suppose first that −n < cont(t⁻¹(p)) −cont((t⁰)⁻¹(p)) < n. If cp , c⁰_p then the hypothesis on the parametercimplies thatk_1−cp−c₀(b_p−a_p),k_1−c⁰_p −c₀(b⁰_p−a⁰_p).

Ifc_p =c_p⁰ then botht⁻¹(p)and(t⁰)⁻¹(p)are addable boxes ofµ^(cp). Since there exists at most one addable box to a Young diagram with a given content, we deduce that the contents oft⁻¹(p)and(t⁰)⁻¹(p)are different, and asc0,0 we havek1−cp−c0(b_p−a_p), k_1−cp −c₀(b⁰_p−a⁰_p).

Therefore we may and will assume that|cont(t⁻¹(p)) −cont((t⁰)⁻¹(p))| ≥ n. Since the absolute value of the content of a box of ad-partition ofncannot exceedn−1, the contents oft⁻¹(p)and of(t⁰)⁻¹(p)are of different signs. Up to exchangingtandt⁰, we suppose that the contentt⁻¹(p)is equal tox>0 and the content of(t⁰)⁻¹(p)is equal to y<0 (neitherxnorycan be equal to 0 since 0≤x <n,−n <y ≤0 andx−y≥n).

The Young diagram[µ]must contain a box of contentx−1 and a box of contenty+1, and therefore has at leastx−y−1 boxes. Sincex−y≥n, we obtain thatp−1≥n−1

and hence p =n. Ifc_p ,c_p⁰ then the d-partition µhas at least x−yboxes, which is impossible, so thatcp =c_p⁰. Butk1−cp −c0x,k1−cp −c0ybecausec0,0.

2.

Leclerc–Miyachi constructible characters

Letqbe an indeterminate overQ. In this section, we introduce other characters ofG(d,1,n), whose definition was given by Leclerc and Miyachi in [10], using canonical bases of some representations over the quantum groupU_q(sl∞). Ifd =2, Leclerc and Miyachi have shown that these characters are equal to Lusztig’s constructible characters [13, Chapter 22]

of the Coxeter group of typeB_n, conjectured to be equal to the Kazhdan–Lusztig cellular characters.

2.1.

The algebra

The quantum groupU_q(sl∞)associated with the doubly infinite Dynkin diagram A∞is theQ(q)-algebra generated byE_i,F_i andK_i^±1fori ∈Zsubject to the following relations:

K_iK_i⁻¹=1=K_i⁻¹K_i, K_iK_j=K_jK_i,

KiEj =q^{−δi,j−1+2δi,j−δi,j+1}EjKi, KiFj =q^{δi,j−1−2δi,j+δi,j+1}FjKi,

E_iF_j−F_jE_i =δ_i,jK_i−K_i⁻¹ q−q⁻¹ , E_iE_j =E_jE_i if|i−j|>1,

E_i²E_j − (q+q⁻¹)E_iE_jE_i+E_jE_i² =0 if|i−j|=1, FiFj =FjFi if|i−j|>1,

F_i²F_j − (q+q⁻¹)F_iF_jF_i+F_jF_i²=0 if|i−j|=1,

for alli,j ∈Z. We can endowU_q(sl∞)with many bialgebra structures, and we choose the following comultiplication∆:

∆(K_i)=K_i ⊗K_i ∆(E_i)=E_i⊗1+K_i⁻¹⊗E_i ∆(E_i)=F_i ⊗K_i+1⊗F_i This bialgebra structure will allow us to consider tensor products of representations of U_q(sl∞).

The fundamental roots ofsl_∞ are denoted by(Λ_i)_i∈Z. For alli ∈ Z, we denote by V(Λ_i)the integrable irreducible representation of highest weightΛ_i. It admits(v_β)_βas aQ(q)-basis, whereβruns in B_i = {(β_k)_k≤i |β_k < β_k+1, β_k =kfork 0}. We will identify such a βin this set with the corresponding subset{β_k |k ≤i}ofZand may write j∈ βorβ∪ {l}ifl<β. On this basis(v_β)_β, the action of the generatorsE_i,F_iand

K_iare [10]:

E_iv_β=

(v_γ ifi< βandi+1∈β, withγ=(β\ {i+1}) ∪ {i}, 0 otherwise,

F_iv_β=

(v_γ ifi∈βandi+1<β, withγ=(β\ {i}) ∪ {i+1}, 0 otherwise,

Kiv_β=













qv_β ifi∈βandi+1<β, q⁻¹v_β ifi< βandi+1∈β, v_β otherwise.

The highest weight vector isv_βi whereβⁱ =Z≤i.

2.2.

Fock spaces, canonical bases and constructible characters

Letr=(r₁, . . . ,rd)be ad-tuple of integers withr1 ≥r2 ≥ · · · ≥rd and we consider the fundamental weightΛ_r=Íd

i=1Λ_ri. The integrable irreducibleU_q(sl_∞)-module of highest weightΛ_ris denoted byV(Λ_r). Denote byF(Λ_r)=V(Λ_r1) ⊗ · · · ⊗V(Λ_rd)the associated Fock space, which admitsv_β^r1⊗ · · · ⊗v_βrd as a highest weight vector of weight Λ_r. From now on, we view the moduleV(Λ_r)inside the Fock spaceF(Λ_r). The module F(Λ_r)has a basisS(Λ_r)given by

S(Λ_r)=

v_β1 ⊗ · · · ⊗v_βd

β_i ∈B_ri .

This is the standard basis ofF(Λ_r)and we prefer to write its indexing set as a set of d-symbols

S=

©

­

­

­

­

­

« β₁ β₂ ... β_d

ª

®

®

®

®

®

¬ ,

whereβ_i =(β_i,k)_k≤ri is a sequence of integers withβ_i,k < β_i,k+1andβ_i,k =kfork0.

The height of such a symbol is the integerÍd i=1Í

k≤ri(β_i,k−k). The highest weight vector v_β^r1 ⊗ · · · ⊗v_βrd of weightΛ_rcorresponds tov_S0, whereS⁰is thed-symbol withi-th line equal toβ^ri. Finally, ad-symbol is said to be standard ifβ_i,k < β_j,k for alli <jand k≤r_j.

Example 2.1. The 3-symbols of height 2 withr=(1,1,0)are the following

©

­

­

«

. . . 0 1 . . . 0 3 . . . 0

ª

®

®

¬

, ©

­

­

«

. . . 0 1 . . . 0 1 . . . 2

ª

®

®

¬

, ©

­

­

«

. . . −1 0 1

. . . −1 0 1

. . . 0 1

ª

®

®

¬

, ©

­

­

«

. . . 0 2 . . . 0 2 . . . 0

ª

®

®

¬

, ©

­

­

«

. . . 0 1 . . . 0 2 . . . 1

ª

®

®

¬ ,

©

­

­

«

. . . 0 3 . . . 0 1 . . . 0

ª

®

®

¬

, ©

­

­

«

. . . 1 2 . . . 0 1 . . . 0

ª

®

®

¬

, ©

­

­

«

. . . 0 1 . . . 1 2 . . . 0

ª

®

®

¬

, ©

­

­

«

. . . 0 2 . . . 0 1 . . . 1

ª

®

®

¬ .

The five first symbols are standard and the four last symbols are not standard.

2.2.2. Canonical bases

Letx7→xbe the involution ofU_q(sl∞)defined as the uniqueQ-linear ring morphism satisfying

q=q⁻¹, K_i =K_i⁻¹ E_i =E_i, F_i =F_i.

SinceV(Λ_r)is a highest weight module with highest weight vectorv_S0, any element v∈V(Λ_r)can be writtenv=xv_S0, withx∈ U_q(sl∞), and we setv=xv_S0.

Let Rbe the subring ofQ(q)of rational functions which are regular atq =0. Let F_R(Λ_r)be theR-sublattice ofF(Λ_r)spanned by the standard basisS(Λ_r).

The canonical basis(b_Σ)_ΣofV(Λ_r)is indexed by the set of standardd-symbols and is characterized by the following properties:

bΣ ≡vΣ modqF_R(Λ_r) and bΣ =bΣ.

Canonical bases were introduced in [12], see also [9]. We will denote this basis byB(Λ_r).

2.2.3. Constructible characters

In [10], a closed expression of anybΣ∈B(Λ_r)in the standard basis is given whend =2, and is compared to Lusztig’s constructible characters. Leclerc and Miyachi then propose a definition of constructible characters via canonical bases for the complex reflection groupG(d,1,n).

To anyd-symbol of heightn, we associate ad-partition(λ⁽¹⁾, . . . , λ^(d))ofnby setting λ⁽_jⁱ⁾=β_i,ri−j+1− (r_i−j+1).

This is a canonical bijection between the set ofd-partitions ofnand the set ofd-symbols of heightn.

Definition 2.2([10, 6.3]). For a standardd-symbolΣof heightn, we write the expression ofbΣin terms of the standard basis

bΣ=Õ

S

a^S_Σ(q)v_S,

with S running in the set of d-partitions of n. The Leclerc–Miyachi r-constructible character ofG(d,1,n)corresponding to the standardd-symbolΣis

γΣ=Õ

S

a^S_Σ(1)χ_S,

where χ_Sis the character of the representation ofG(d,1,n)associated with the partition corresponding to thed-symbolS.

2.3.

The asymptotic case

Since we aim to compare the set of Calogero–Moser cellular characters and the set of Leclerc–Miyachi constructible character, we expect that the Leclerc–Miyachi constructible characters enjoy a generic property similar to Corollary 1.11. This generic property on the parametercwill turn out to be an asymptotic property on the parameterr.

Lemma 2.3. Let r = (r₁, . . . ,r_d)and n ∈ N. We suppose thatr_i −r_i+1 ≥ n for all 1≤i ≤d−1. Then everyd-symbol of height at mostnis standard.

Proof. LetS=(β_i)_1≤i≤dbe a symbol of height smaller thann. By the hypothesis on the parameters, we necessarily have β_i,k =kfork ≤r_i+1. Therefore ifi< jandk≤rjwe haveβ_i,k =k≤ β_j,k and thed-symbolSis standard.

If r_i −r_i+1 ≥ n then the number of r-constructible characters ofG(d,1,n)is the same as the number of irreducible characters ofG(d,1,n). It remains to show that these r-constructible characters are irreducible.

Theorem 2.4. Letr =(r₁, . . . ,r_d)andn ∈ N. We suppose thatr_i −r_i+1 ≥ n for all 1≤i ≤d−1. For anyd-symbolΣof height at mostnwe haveb_Σ=v_Σ.

Proof. This is an application of the algorithm presented in [11] for the computation of the canonical basis. We show that the intermediate basis(A_Σ)_Σof [11, Section 4.1] satisfies

AΣ=vΣ, which implies thatbΣ =vΣsinceAΣ =AΣ. We proceed by induction onn.

Forn=1, anyd-symbol of height 1 is given bySl=(β_i)_1≤i≤dwithβ_i,k =kfor every 1≤i ≤dandk ≤r_i except forβ_l,rl =r_l+1. We immediately obtain thatA_Sl =F_rlv_S0. By the hypothesis onr, the only lineβ_k ofΣwithrl ∈ β_k andrl+1<β_k isβ_l. Therefore F_rlv_S0 =v_Sl.

Suppose that for all parametersrsuch thatr_i−r_i+1 ≥nfor all 1≤i ≤d we have AΣ = v_Σ for all standard d-symbols Σ of heightn. Let rbe a parameter such that r_i−r_i+1 ≥n+1 for all 1≤i ≤dandΣ=(β_i)_1≤i≤d be a standardd-symbol of height n+1. Leti0be the greatest integer such thatβ_i0 ,β^ri0 andk0the smallest integer such that β_i0,k0 >k₀. We write β_i0,k0 =k₀⁰+1 withk₀⁰ ≥ k₀. Since the height ofΣisn+1, we have β_i0,k0−k₀ ≤ n+1 so that k₀⁰ ≤ n+k₀. In order to apply the algorithm of Leclerc–Toffin, one must find the smallest integerksuch that there exists 1≤i≤dand l≤kwithβ_i,l =k+1. Let us show that this integer isk₀⁰.

Fix k < k₀⁰, 1 ≤i ≤d andl ≤ k. Suppose first thati >i0. By definition ofi0, we have β_i,l =l ,k+1. Now suppose thati =i₀. Ifl < k₀ then by definition ofk₀ we have β_i0,l =l ,k+1. Ifl ≥k0 then β_i0,l ≥ β_i0,k0 =k₀⁰+1>k+1. Finally, suppose thati<i₀. Sincel ≤k, we havel ≤k₀⁰+1≤k₀+n+1. SinceΣis of heightn+1, if l≤r_i− (n+1− (β_i0,k0−k₀))we haveβ_i,l =l. Butr_i− (n+1) ≥r_i+1 ≥r_i0 and therefore r_i − (n+1− (β_i0,k0 −k₀)) ≥ r_i0 +k₀⁰+1−k₀. As obviouslyk₀ ≤ r_i0 we obtain that r_i− (n+1− (β_i0,k0−k₀)) ≥k₀⁰+1 ≥k. Hence ifl ≤kthenβ_i,l =l ,k+1.

Therefore we obtain

AΣ=Fk₀⁰AΣ⁰,

whereΣ⁰is the standardd-symbol obtained fromΣby replacing onlyβ_i0,k0byk₀⁰. Then Σ⁰is of heightn, and the induction hypothesis shows thatAΣ⁰=vΣ⁰.

In order to conclude, it remains to show thatF_k⁰

0vΣ⁰=vΣ. Ifi>i0thenβ_i =β^riand sincek₀⁰ ≥k₀>r_ineitherk₀nork₀⁰appear in β_i. Ifi <i₀, we have already shown that if l≤ri− (n+1− (β_i0,k0−k0))we haveβ_i,l =land thatri− (n+1− (β_i0,k0−k0)) ≥k₀⁰+1 so thatβ_i,k⁰

0+1 =k₀⁰+1 andβ_i,k⁰

0 =k₀⁰and bothk₀⁰andk₀⁰+1 appear inβ_i. Hence, from the definition of the action ofF_k⁰

0via the comultiplication, we find thatF_k⁰

0vΣ⁰=vΣ. The following corollary translates Theorem 2.4 in terms of Leclerc–Miyachi con- structible characters forG(d,1,n).

Corollary 2.5. Letr=(r₁, . . . ,rd)andn ∈ Nand suppose thatri −r_i+1 ≥ nfor all 1≤i≤d−1. Then the Leclerc–Miyachir-constructible characters for the groupG(d,1,n) are the irreducible characters.

3.

Computations for

and comparison

In this section, we compute explicitly the set of c-cellular characters for the group G(d,1,2)for any choice of parameterc. We also compute explicitly the Leclerc–Miyachi r-constructible characters for any choice of parameterr.

3.1.

On the Calogero–Moser side

We will freely use the notations of Section 1, but simplify them in the special case of G(d,1,2). For simplicity, we prefer to denote by(x,y)the standard basis ofC²and by (X,Y)its dual basis. Letsbe the reflection denoted bys_1,2,0andtbe the reflection denoted byσ₁, so thatG(d,1,2)has the following presentation

s,t

s²=1,t^d =1,stst=tsts. We also denote bys_k the reflections_1,2,k. There are 2d+ ^d₂

irreducible representations ofG(d,1,2), namelyη_i, η_i⁰of dimension 1 for 1 ≤i ≤ d andρ_i,j of dimension 2 for 1≤i< j ≤d. Their respective characters are denoted byξ_i, ξ_i⁰and χ_i,j and the values of the representations on the generators are given in Table 3.1

Table 3.1. Action ofsandton irreducible repesentations ofG(d,1,2)

s t

η_i,1≤i≤d 1 ζⁱ⁻¹

η_i⁰,1≤i ≤d −1 ζⁱ⁻¹

ρ_i,j,1≤i< j ≤d

0 1 1 0

ζ^i−j 0 0 ζ^j−i

Finally we choose a parameterc: Ref(G(d,1,2)) →C, definek_i^#=k_1−iand we again setζ =ζ_d.

The Gaudin algebra GaucoverC[X,Y]is generated by the following two elements D_x=

d−1

Õ

k=0

c_kζ^k1 Xσ₁^k−c₀

d−1

Õ

k=0

1

X−ζ^kYs_k and D_y=

d−1

Õ

k=0

c_kζ^k1 Yσ₂^k+c₀

d−1

Õ

k=0

ζ^k X−ζ^kYs_k. SinceC(X,Y)Gauc⊂C(V)G(d,1,2), every irreducibleC(X,Y)Gauc-module appears in the restriction of an irreducible representation ofG(d,1,2)overC(X,Y). We then denote byL_i (resp.L⁰

i, resp.L_i,j) the restriction ofC(V)η_i (resp.C(V)η_i⁰, resp.C(V)ρ_i,j) to C(X,Y)Gauc. The following easy lemma will be useful in the computations.

Lemma 3.1. InC(V)=C(X,Y), for every1≤l ≤dwe have

d−1

Õ

k=0

ζ^kl

X−ζ^kY = dX^l−1Y^d−l

X^d−Y^d . (3.1)

We also give two other generators ofC(X,Y)Gauc, which differ fromD_xandD_yby multiplication by a scalar:

D_x⁰ = X(X^d−Y^d)

d D_x and D_y⁰ =Y(X^d−Y^d) d D_y.