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

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 level $d$ irreducible integrable representations of $\mathcal{U}_q(\mathfrak{sl}_{\infty})$. We prove this conjecture in some cases: in full generality for $G(d,1,2)$ and for generic parameters for $G(d,1,n)$.

Using Cherednik algebras and Calogero-Moser spaces, Bonnafé and Rouquier developed in [4] the notions of cells (right, left or two-sided) and of cellular characters of a complex reflection group W . 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 parameter c defined on the reflections of W and invariant by conjugation. In this paper, we are mainly interested in the notion of cellular characters for the complex reflection group Gpd, 1, nq. If d " 1, the group Gpd, 1, nq is nothing else than the Weyl group of type A n´1 , and if d " 2, we recover the Weyl group of type B n . Bonnafé and Rouquier showed that if the Calogero-Moser space with parameter c associated 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 type A. Even in type B, we only have a complete description for B 2 [4,Chapter 19]. For the dihedral group Gpd, d, 2q, 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 conjectures 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 representation V pΛ r 1`Λ r 2 q of U q psl 8 q (here the Λ i are the fundamental weights). By evaluating these expressions at q " 1, Leclerc and Miyachi retrieved Lusztig's constructible characters for Weyl groups of type B and D. With a level d irreducible integral representation V´ř d i"1 Λ r i¯, they defined some characters of the complex reflection group Gpd, 1, nq in a similar manner and asked whether these characters are a good analogue of constructible characters in type B.
Thus, we have two sets of characters for the complex reflection group Gpd, 1, nq, namely the Calogero-Moser cellular characters and the constructible characters of Leclerc and Miyachi. Both sets of characters depend heavily on some parameters (c or r " pr 1 , . . . , r d q), and up to a suitable change of parameters, we conjecture that these two sets of characters are equal.
Conjecture A (Conjecture 4.1). Let r be a d-tuple of integers. Then there exists an explicit choice of parameter c for the complex reflection group Gpd, 1, nq such that the set of Calogero-Moser c-cellular characters and the set of Leclerc-Miyachi r-constructible characters coincide.
We refer to the statement in Section 4 for the precise relation between the parameters c and r. The main result of this paper is a proof of this conjecture in two different cases.
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 minimal b-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 Gpd, 1, nq 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 group Gpd, 1, 2q. On the Calogero-Moser side, we diagonalize the action of the Gaudin algebra on representations of Gpd, 1, 2q 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 for Gpd, 1, 2q and any choice of parameters, and for Gpd, 1, nq for generic parameters.

Calogero-Moser cellular characters
We introduce the set of Calogero-Moser c-cellular characters of a complex reflection group, which we define using the notion of Gaudin algebra, see [4]. In the specific case of Gpd, 1, nq, we introduce a commutative subalgebra JM c generated by the so-called Jucys-Murphy elements. Using results of the Appendix, we show that the cellular characters of Gpd, 1, nq for the algebra JM c are sums of Calogero-Moser characters. For specific values of c, we show that the cellular characters of Gpd, 1, nq for the algebra JM c are irreducible, then so are the Calogero-Moser c-cellular characters.
1.1. Notations. We fix V a finite dimensional C-vector space, denote by det : GLpV q Ñ C˚the determinant and by x¨,¨y : VˆV˚Ñ C the duality between V and the space V˚of linear forms on V . We choose for each positive integer d a d-th root of unity ζ d such that ζ d{l d " ζ l for all l dividing d. The group of d-th roots of unity will be denoted by µ d .
Let W Ă GLpV q be a finite complex reflection group. We denote by RefpW q the set of pseudo-reflections of W and for each s P RefpW q, we choose α s P V˚and α _ s P V such that kerps´Id V q " kerpα s q and Imps´Id V q " Cα _ s . We denote by A the set of reflecting hyperplanes of W as well as by V reg the open subset V z Ť HPA H. A theorem of Steinberg [6,Theorem 4.7] shows that V reg is the subset of elements of V with trivial stabilizers with respect to the action of W .
For H P A, the pointwise stabilizer W H of H is a cyclic group of order e H with a chosen generator s H P RefpW q. If Ω P A{W , we denote by e Ω the common value of e H for H P Ω. With these notations, the set of reflections of W is RefpW q " s j HˇH P A, 1 ď j ď e H´1 ( , and two reflections s j H and s j 1 H 1 are conjugate if and only if the hyperplanes H and H 1 are in the same orbit under the action of W and j " j 1 .
We also fix c : RefpW q Ñ C, s Þ Ñ c s a function which is invariant by conjugation. For any H P A and 0 ď i ď e H´1 , we define which satisfy ř e H i"0 k H,i " 0; we will often consider indices modulo e H and set k Ω,i " k H,i for Ω P A{W and any H P Ω. We recover the function c via 1.2. Gaudin algebra and Calogero-Moser cellular characters. For any y P V , we define an element D y in the group ring of W with coefficients in CrV reg s: The Gaudin algebra Gau c pW q is the sub-CrV s-algebra of CrV reg sW generated by p D y q yPV . 4, 13.4.B]). The algebra Gau c pW q is commutative.
Therefore we are in the situation of the Appendix if we set E " CW , A " CW acting on E by left multiplication, P " CrV reg s and D i " D y i acting on E by right multiplication, where py i q i is a basis of V . The following is Definition A.1 in our setting.
Definition 1.2. The set of cellular characters for the algebra Gau c pW q is called the set of Calogero-Moser c-cellular characters, or for short c-cellular characters. The c-cellular character associated to L P IrrpCpV q Gau c pW qq is where V χ is a representation of W affording the character χ and rX : Ls is the multiplicity of L in the module X. For all z P V reg , we can specialize D y to an element of CW , by evaluating the coefficents in CrV reg s at z: ÿ sPRefpW q c s detpsq xα s , yy xα s , zy s P CW.
Choosing z " y, one obtains a central element of CW , called the Euler element, which does not depend on y: For Ω P A{W and H P Ω, we define m j Ω,χ " xχ |W H , det´j |W H y W H , where x¨,¨y W H denotes the scalar product of characters of W H . The Euler element eu c acts on a representation affording the character χ by multiplication by k Ω,j .
1.3. The imprimitive reflection group Gpd, 1, nq. In this subsection, V is of dimension n with a chosen basis py 1 , . . . , y n q with dual basis px 1 , . . . , x n q. Using this choice of basis, we identify GLpV q to GL n pCq. We also fix a positive integer d and denote by ζ the d-th root of unity ζ d .
1.3.1. The group Gpd, 1, nq and its reflections. It is easy to describe the group Gpd, 1, nq in terms of matrices: it is the subgroup of GLpCq with elements the monomial matrices with coefficients in µ d . The permutation matrix corresponding to the transposition pi jq will be denoted by s i,j and the diagonal matrix with diagonal entries p1, . . . , 1, ζ, 1, . . . , 1q, with ζ at the i-th position will be denoted by σ i . The set of reflections of Gpd, 1, nq splits into d conjugacy classes: where RefpGpd, 1, nqq 0 " σ r i s i,j σ´r iˇ1 ď i ă j ď n, 0 ď r ď d´1 for 1 ď k ď d´1. We now give an explicit choice for α s and α _ s for any reflection s. For the reflection s i,j,r " σ r i s i,j σ´r i , the reflecting hyperplane H i,j,r is given by the kernel of the linear form α i,j,r " x i´ζ r x j and the eigenspace associated to the eigenvalue´1 is spanned by α _ i,j,r " ζ r y i´yj . For the reflection σ k i , the reflecting hyperplane H i is given by the kernel of the linear form α i " x i and the eigenspace associated to the eigenvalue ζ k is spanned by Under the action of Gpd, 1, nq, the set of reflecting hyperplanes A has only two orbits, Ω 0 " tH i,j,r | 1 ď i ă j ď n, 0 ď r ă d´1u and Ω 1 " tH i | 1 ď i ď nu which are of respective cardinal d npn´1q 2 and n. Given a function c : RefpW q Ñ C constant on the conjugacy classes, we denote its value on RefpGpd, 1, nqq k by c k and will write k i instead of k Ω 1 ,i . We will try not to introduce the parameters k Ω 0 ,0 and k Ω 0 ,1 which are respectively equal to´c 0 2 and c 0 2 . Finally, the Euler element associated to Gpd, 1, nq and c will be denoted by eu c,n .

1.3.2.
Representations and d-partitions. The representation theory of Gpd, 1, nq is well known and is governed by the d-partitions of n, see [8, Section 5.1] for example. A partition of n is a finite sequence of integers λ " pλ 1 , . . . , λ r q adding up to n such that λ 1 ě λ 2 ě¨¨¨λ r ą 0, and we set |λ| " n. A d-partition of n is a d-tuple pλ p1q , . . . , λ pdq q of partitions such that ř d i"1 |λ piq | " n. The isomorphism classes of irreducible complex representations of Gpd, 1, nq are parameterized by d-partitions of n, and for such a d-partition λ, we denote by V λ a corresponding representation.
One can describe the branching rule Gpd, 1, nq Ă Gpd, 1, n`1q in terms of Young diagrams. The Young diagram rλs of a d-partition λ of n is the set , whose elements will be called boxes. The content contpγq of a box γ " pa, b, cq is the integer b´a. A box γ of rλs is said to be removable if rλsztγu is the Young diagram of a d-partition µ of n´1, and in this case, the box γ is said to be addable to µ. Gpd,1,nq pV λ q " where µ runs over the d-partitions of n`1 with Young diagram obtained by adding an addable box to the Young diagram of λ. Concerning the restriction,

Res
Gpd,1,nq where µ runs over the d-partitions of n´1 with Young diagram obtained by removing a removable box from the Young diagram of λ.
Using this branching rule, we define a basis of V λ in terms of standard d-tableaux of shape λ, which are bijections t : rλs Ñ t1, . . . , nu such that for all boxes γ " pa, b, cq and γ 1 " pa 1 , b 1 , cq we have tpγq ă tpγ 1 q if a " a 1 and b ă b 1 or a ă a 1 and b " b 1 . Giving a standard d-tableau is then equivalent to giving a sequence of d-partitions pλ t risq 1ďiďn such that rλ t riss " t´1pt1, . . . , iuq. Therefore V λ is the direct sum of one dimensional spaces D t , where for all 1 ď i ď n the space D t is in the irreducible component V λ t ris of Res Gpd,1,nq Gpd,1,iq pV λ q.

1.3.3.
A commutative subalgebra of CGpd, 1, nq. For 1 ď k ď n, we define the following elements of CGpd, 1, nq: If d " 1, these elements are the usual Jucys-Murphy elements for the symmetric group S k , multiplied by the scalar c 0 . Lemma 1.6. For all 1 ď i, j ď n, the Jucys-Murphy elements J i and J j commute and It is immediate to check that the Jucys-Murphy element J i is equal to Since the conjugate of s p,i,r by s i,i`1,0 is s p,i`1,r and the conjugate of σ r i by s i,i`1,0 is σ r i`1 , we have that and we obtain the induction formula.
Since eu c,i commutes with every element of CGpd, 1, iq, one see that J i " eu c,i´e u c,i´1 commutes with every element of CGpd, 1, i´1q, and therefore with J 1 , . . . , J i´1 .
The commutative subalgebra of CGpd, 1, nq generated by the elements J 1 , . . . , J n is denoted by JM c pd, nq. On the representation V 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. Let λ be a d-partition of n and γ " pa, b, cq a removable box of rλs. Then J n acts on the component V λztγu of Res Gpd,1,nq Gpd,1,n´1q pV λ q by multiplication by the scalar dpk 1´c´c0 pb´aqq.
Proof. We start by computing the action of the Euler element on V λ , and therefore we need the values of the integers m χ λ Ω,j , for Ω P A{W and 0 ď j ď e e Ω´1 , where χ λ is the character of V λ . From [14, Lemma 6.1] we have Therefore, using Lemma 1.4, eu c,n acts on V λ by multiplication by the scalar But xpχ λ q | xs 1 y , 1y xs 1 y " χ λ p1q´xpχ λ q | xs 1 y , dety xs 1 y , so that and we obtain the desired formula because J n " eu c,n´e u c,n´1 .
Corollary 1.8. Let λ be a d-partition of n, t be a standard d-tableau of shape λ and 1 ď k ď n. The element J k acts on D t by multiplication by dpk 1´c´c0 pb´aqq, where t´1pkq " pa, b, cq.

1.3.4.
Cellular characters for JM c pd, nq and c-cellular characters. As well as for the Gaudin algebra, we define cellular characters for the algebra JM c pd, nq. We are again in the situation of the Appendix if we set E " CGpd, 1, nq, A " CGpd, 1, nq acting on E by left multiplication, P " C and D i " J i acting on E by right multiplication. The following is Definition A.1 in our setting.
where V χ is a representation of W affording the character χ.
These characters are easier to compute than the c-cellular characters, since JM c pd, nq is commutative and split, and are close to c-cellular characters in the following sense.
Theorem 1.10. Every cellular character for the algebra JM c pd, nq is a sum of c-cellular characters.
Proof. We denote by Gau p0q c pd, nq the sub-CrV s-algebra of CrV reg sGpd, 1, nq generated by for 1 ď k ď n. Since x k is invertible in CpV q for all k, the algebras CpV q Gau c pGpd, 1, nqq and CpV q Gau p0q c pd, nq are equal, and the c-cellular characters are therefore equal to the cellular characters for Gau p0q c pd, nq. We then specialize the algebra Gau p0q c pd, nq with respect to the following increasing sequence of prime ideals and we denote by π piq : Gau piq c pd, nq Ñ Gau pi`1q c pd, nq the corresponding quotient map, and Π piq " π pi´1q˝¨¨¨πp0q . The algebra Gau pnq c pd, nq is hence a subalgebra of CGpd, 1, nq. By Proposition A.3, the cellular characters for the algebra Gau piq c pd, nq are sums of cellular characters for the algebra Gau pi´1q c pd, nq, and therefore the cellular characters for the algebra Gau piq c pd, nq are sums of c-cellular characters. An easy induction shows that if i ě k then The algebra Gau pnq c pd, nq is thus equal to JM c pd, nq since Π pnq px k D k q " J k , which ends the proof.
Corollary 1.11. Suppose that the parameter c is such that c 0 ‰ 0 and pk p´kq q´c 0 j ‰ 0, for all 1 ď p ‰ q ď d and´n ă j ă n. Then the Calogero-Moser c-cellular characters of Gpd, 1, nq are irreducible.
Proof. Since JM c pd, nq Ă CGpd, 1, nq, any simple representation occurs in some V λ and is therefore of the form D t for t a standard d-tableau. We show that these one dimensional representations of JM c pn, dq are pairwise non-isomorphic. Let t and t 1 be two distinct standard d-tableaux. We prove that the sequences pk 1´cp´c0 pb p´ap qq 1ďpďn and pk 1´c 1 p´c 0 pb 1 are different, where pa p , b p , c p q " t´1ppq and pa 1 p , b 1 p , c 1 p q " pt 1 q´1ppq. Let 1 ď p ď n be the minimal integer such that t´1ppq and pt 1 q´1ppq are different. Denote by µ the common partition λ t rp´1s " λ t 1 rp´1s.
Suppose first that´n ă contpt´1ppqq´contppt 1 q´1ppqq ă n. If c p ‰ c 1 p then the hypothesis on the parameter c implies that k 1´cp´c0 pb p´ap q ‰ k 1´c 1 p´c 0 pb 1 p´a 1 p q. If c p " c 1 p then both t´1ppq and pt 1 q´1ppq are addable boxes of µ pcpq . Since there exists at most one addable box to a Young diagram with a given content, we deduce that the contents of t´1ppq and pt 1 q´1ppq are different, and as c 0 ‰ 0 we have k 1´cp´c0 pb p´ap q ‰ k 1´cp´c0 pb 1 p´a 1 p q. Therefore we may and will assume that |contpt´1ppqq´contppt 1 q´1ppqq| ě n. Since the absolute value of the content of a box of a d-partition of n cannot exceed n´1, the contents of t´1ppq and of pt 1 q´1ppq are of different signs. Up to exchanging t and t 1 , we suppose that the content t´1ppq is equal to x ą 0 and the content of pt 1 q´1ppq is equal to y ă 0 (neither x nor y can be equal to 0 since 0 ď x ă n,´n ă y ď 0 and x´y ě n). The Young diagram rµs must contain a box of content x´1 and a box of content y`1, and therefore has at least x´y´1 boxes. Since x´y ě n, we obtain that p´1 ě n´1 and hence p " n. If c p ‰ c 1 p then the d-partition µ has at least x´y boxes, which is impossible, so that c p " c 1 p . But k 1´cp´c0 x ‰ k 1´cp´c0 y because c 0 ‰ 0.

Leclerc-Miyachi constructible characters
In this section, we introduce other characters of Gpd, 1, nq, whose definition was given by Leclerc and Miyachi in [10], using canonical bases of some representations over the quantum group U q psl 8 q. If d " 2, Leclerc and Miyachi have shown that these characters are equal to Lusztig's constructible characters [13,Chapter 22] of the Coxeter group of type B n , conjectured to be equal to the Kazhdan-Lusztig cellular characters. Let q be an indeterminate over Q.
2.1. The Hopf algebra U q psl 8 q. The quantum group U q psl 8 q associated with the doubly infinite Dynkin diagram A 8 is the Qpqq-algebra generated by E i , F i and K˘1 i for i P Z subject to the following relations: It is a Hopf algebra, and we choose the following comultiplication ∆, counit ε and antipode S: The fundamental roots of sl 8 are denoted by pΛ i q iPZ . For all i P Z, we denote by V pΛ i q the integrable irreducible representation of highest weight Λ i . It admits pv β q β as a Qpqqbasis, where β runs in tpβ k q kďi | β k ă β k`1 , β k " k for k ! 0u. We will identify such a β in this set with the corresponding subset tβ k | k ď iu of Z and may write j P β or β Y tlu if l R β. On this basis pv β q β , the action of the generators E i , F i and K i are [10]: The highest weight vector is v β i where β i " Z ďi .
2.2.1. Fock space of a representation. Let r " pr 1 , . . . , r d q be a d-tuple of integers with r 1 ě r 2 ě¨¨¨ě r d and we consider the fundamental weight Λ r " ř d i"1 Λ r i . The integrable irreducible U q psl 8 q-module of highest weight Λ r is denoted by V pΛ r q. Denote by F pΛ r q " V pΛ r 1 q b¨¨¨b V pΛ r d q the associated Fock space, which admits v β r 1 b¨¨¨b v β r d as a highest weight vector of weight Λ r . From now on, we view the module V pΛ r q inside the Fock space F pΛ r q. The module F pΛ r q has a basis SpΛ r q given by This is the standard basis of F pΛ r q and we prefer to write its indexing set as a set of d-symbols where β i " pβ i,k q kďr i is a sequence of integers with β i,k ă β i,k`1 and β i,k " k for k ! 0.
The height of such a symbol is the integer Finally, a d-symbol is said to be standard if β i,k ă β j,k for all i ď j and k ď r j .

2.2.2.
Canonical bases. Let x Þ Ñ x be the involution of U q psl 8 q defined as the unique Q-linear ring morphism satisfying Since V pΛ r q is a highest weight module with highest weight vector v S 0 , any element v P V pΛ r q can be written v " xv S 0 , with x P U q psl 8 q, and we set v " xv S 0 . Let R be the subring of Qpqq of rational functions which are regular at q " 0. Let F R pΛ r q be the R-sublattice of F pΛ r q spanned by the standard basis SpΛ r q.
The canonical basis pb Σ q Σ of V pΛ r q is indexed by the set of standard d-symbols and is characterized by the following properties: Canonical bases were introduced in [12], see also [9]. We will denote this basis by BpΛ r q.

Constructible characters.
In [10], a closed expression of any b Σ P BpΛ r q in the standard basis is given when d " 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 group Gpd, 1, nq.
To any d-symbol, we associate a d-partition pλ p1q , . . . , λ pdq q of its height by setting λ piq j " β i,r i´j`1´p r i´j`1 q. This is a bijection between the set of d-partitions of n and the set of d-symbols of height n.
where χ S is the character of the representation of Gpd, 1, nq associated with the partition corresponding to the d-symbol S.
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 parameter c will turn out to be an asymptotic property on the parameter r.
Lemma 2.2. Let r " pr 1 , . . . , r d q and n P N. We suppose that r i´ri`1 ě n for all 1 ď i ď d´1. Then every d-symbol of height at most n is standard.
Proof. Let S " pβ i q 1ďiďd be a symbol of height smaller than n. By the hypothesis on the parameters, we necessarily have β i,k " k for k ď r i`1 . Therefore if i ă j and k ď r j we have β i,k " k ď β j,k and the d-symbol S is standard.
If r i´ri`1 ě n then the number of r-constructible characters of Gpd, 1, nq is the same as the number of irreducible characters of Gpd, 1, nq. It remains to show that these rconstructible characters are irreducible. Theorem 2.3. Let r " pr 1 , . . . , r d q and k P N. We suppose that r i´ri`1 ě n for all 1 ď i ď d´1. For any d-symbol Σ of height at most n we have b 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 pA Σ q Σ of [11, Section 4.1] satisfies A Σ " v Σ , which implies that b Σ " v Σ since A Σ " A Σ . We proceed by induction on n.
For n " 1, any d-symbol of height 1 is given by S l " pβ i q 1ďiďd with β i,k " k for every 1 ď i ď d and k ď r i except for β l,r l " r l`1 . We immediately obtain that A S l " F r l v S 0 . By the hypothesis on r, the only line β k of Σ with r l P β k and r l`1 R β k is β l . Therefore Suppose that for all parameters r such that r i´ri`1 ě n for all 1 ď i ď d we have A Σ " v Σ for all standard d-symbols Σ of height n. Let r be a parameter such that r i´ri`1 ě n`1 for all 1 ď i ď d and Σ " pβ i q 1ďiďd be a standard d-symbol of height n`1. Let i 0 be the greatest integer such that β i 0 ‰ β r i 0 and k 0 the smallest integer such that β i 0 ,k 0 ą k 0 . We write β i 0 ,k 0 " k 1 0`1 with k 1 0 ě k 0 . Since the height of Σ is n`1, we have β i 0 ,k 0´k 0 ď n`1 so that k 1 0 ď n`k 0 . In order to apply the algorithm of Leclerc-Toffin, one must find the smallest integer k such that there exists 1 ď i ď d and l ď k with β i,l " k`1. Let us show that this integer is k 1 0 . Fix k ă k 1 0 , 1 ď i ď d and l ď k. Suppose first that i ą i 0 . By definition of i 0 , we have β i,l " l ‰ k`1. Now suppose that i " i 0 . If l ă k 0 then by definition of k 0 we have β i 0 ,l " l ‰ k`1. If l ě k 0 then β i 0 ,l ě β i 0 ,k 0 " k 1 0`1 ą k`1. Finally, suppose that i ă i 0 . Since l ď k, we have l ď k 1 0`1 ď k 0`n`1 . Since Σ is of height n`1, if l ď r i´p n`1´pβ i 0 ,k 0´k 0 qq we have β i,l " l. But r i´p n`1q ě r i`1 ě r i 0 and therefore r i´p n`1´pβ i 0 ,k 0´k 0 qq ě r i 0`k 1 0`1´k 0 . As obviously k 0 ď r i 0 we obtain that r i´p n`1´pβ i 0 ,k 0´k 0 qq ě k 1 0`1 ě k. Hence if l ď k then β i,l " l ‰ k`1. Therefore we obtain A Σ " F k 1 0 A Σ 1 , where Σ 1 is the standard d-symbol obtained from Σ by replacing only β i 0 ,k 0 by k 1 0 . Then Σ 1 is of height n, and the induction hypothesis shows that A Σ 1 " v Σ 1 .
In order to conclude, it remains to show that F k 1 0 v Σ 1 " v Σ . If i ą i 0 then β i " β r i and since k 1 0 ě k 0 ą r i neither k 0 nor k 1 0 appear in β i . If i ă i 0 , we have already shown that if l ď r i´p n`1´pβ i 0 ,k 0´k 0 qq we have β i,l " l and that r i´p n`1´pβ i 0 ,k 0´k 0 qq ě k 1 0`1 so that β i,k 1 0`1 " k 1 0`1 and β i,k 1 0 " k 1 0 and both k 1 0 and k 1 0`1 appear in β i . Hence, from the definition of the action of F k 1 0 via the comultiplication, we find that F k 1 0 v Σ 1 " v Σ . The following corollary translates Theorem 2.3 in terms of Leclerc-Miyachi constructible characters for Gpd, 1, nq.
Corollary 2.4. Let r " pr 1 , . . . , r d q and k P N and suppose that r i´ri`1 ě n for all 1 ď i ď d´1. Then the Leclerc-Miyachi r-constructible characters for the group Gpd, 1, nq are the irreducible characters.

Computations for Gpd, 1, 2q and comparison
In this section, we compute explicitly the set of c-cellular characters for the group Gpd, 1, 2q for any choice of parameter c. We also compute explicitly the Leclerc-Miyachi r-constructible characters for any choice of parameter r.

3.1.
On the Calogero-Moser side. We will freely use the notations of Section 1, but simplify them in the special case of Gpd, 1, 2q. For simplicity, we prefer to denote by px, yq the standard basis of C 2 and by pX, Y q its dual basis. Let s be the reflection denoted by s 1,2,0 and t be the reflection denoted by σ 1 , so that Gpd, 1, 2q has the following presentation @ s, tˇˇs 2 " 1, t d " 1, stst " tsts D .
We also denote by s k the reflection s 1,2,k . There are 2d``d 2˘i rreducible representations of Gpd, 1, 2q, namely η i , η 1 i of dimension 1 for 1 ď i ď d and ρ i,j of dimension 2 for 1 ď i ă j ď d. Their respective characters are denoted by ξ i , ξ 1 i and χ i,j and the values of the representations on the generators are given in Table 1 s Action of s and t on irreducibles repesentations of Gpd, 1, 2q Finally we choose a parameter c : RefpGpd, 1, 2qq Ñ C, define k # i " k 1´i and we again set ζ " ζ d .
The Gaudin algebra Gau c over CrX, Y s is generated by the following two elements Since CpX, Y q Gau c Ă CpV qGpd, 1, 2q, every irreducible CpX, Y q Gau c -module appears in the restriction of an irreducible representation of Gpd, 1, 2q over CpX, Y q. We then denote by L i (resp. L 1 i , resp. L i,j ) the restriction of CpV qη i (resp. CpV qη 1 i , resp. CpV qρ i,j ) to CpX, Y q Gau c . The following easy lemma will be useful in the computations.
Lemma 3.1. In CpV q " CpX, Y q, for every 1 ď l ď d we have We also give two other generators of CpX, Y q Gau c , which differ from D x and D y by multiplication by a scalar:  Table 2.
Proof. Lets us start with the action of D 1 x on L i . It is given by the last equality following from the definition k # i and from Lemma 3.1. Similar computations can be made for the action of D 1 y , and for the representation L 1 i .
Let 1 ď i ă j ď n and we compute the action of D 1 x on L i,j : using again the definition of k # i and k # j and Lemma 3.1. The action of D 1 y is obtained by a similar argument.
The 2-dimensional representations L i,j have different behaviour depending on the parameter c.
3.1.1. When c 0 " 0. In this subsection only, we suppose that c 0 " 0. The matrices giving the action of D 1 x and D 1 y are all diagonal and we readily see that L i » L 1 i and that L i,j »L i,j 'L j,i , whereL i,j is the 1-dimensional representation where D 1 x acts by pX d´Y d qk # i and D 1 y by pX d´Y d qk # j . Notice that with this notation, the moduleL i,i is nothing else than L i . Moreover, we have an isomorphism betweenL i,j andL p,q if and only if k # i " k # p and k # j " k # q . We therefore define an equivalence relation on the set t1, . . . , du by i " j if and only if k # i " k # j . Simple CpX, Y q Gau c -modules are then parameterized by pairs of equivalence class for ": a representative of the class L O,O 1 labeled by O and O 1 isL i,j , where i P O and j P O 1 .
From the above description of restrictions of representations of Gpd, 1, 2q to CpX, Y q Gau c , we obtain: The c-cellular character corresponding to the class L O,O 1 is By definition, every c-cellular character is equal to one of those above.

3.1.2.
When c 0 ‰ 0. In this subsection only, we suppose that c 0 ‰ 0. The matrices giving the action of D 1 x and D 1 y on the representation L i,j are not diagonal, but these representations can still have an invariant one-dimensional subspace.
i " k # j and d is even then L i,j is isomorphic to Lì ,j ' Lí ,j , where Lì ,j and Lí ,j are two non-isomorphic one-dimensional representations, which are not isomorphic to some L k or L 1 k , We diagonalize the matrices ρ i,j p D 1 x q and ρ i,j p D 1 y q. Note that these two matrices have the same trace and determinant, and therefore the same characteristic polynomial equal to This homogeneous polynomial is then a square in CpX, Y q if and only if d is even and k re common eigenvectors for ρ i,j p D 1 x q and ρ i,j p D 1 y q with respective eigenvalues pX d´Y d qk # i`c 0 X d{2 Y d{2 and pX d´Y d qk # i´c 0 X d{2 Y d{2 for ρ i,j p D 1 x q and pX d´Y d qk # i´c 0 X d{2 Y d{2 and pX d´Y d qk # i`c 0 X d{2 Y d{2 for ρ i,j p D 1 y q. This shows the first assertion of the lemma. Now, suppose that k # i´k # j " c 0 . We check that Y j´i X j´i˙a ndˆX d´pj´iq Y d´pj´iqȧ re common eigenvectors for ρ i,j p D 1 x q and ρ i,j p D 1 y q with respective eigenvalues y q. Moreover, using the equality c 0 " k # i´k # j , it is easy to see that this gives an isomorphism L i,j » L i ' L 1 j . If k # i´k # j "´c 0 , a similar argument shows that L i,j » L 1 i ' L j , which ends the proof of the lemma.
We now have a complete description of simple CpX, Y q Gau c -modules, and of the restrictions of the representations of Gpd, 1, 2q to CpX, Y q Gau c . The only isomorphism between simple modules are the following: Lì ,j » Lp ,q and Lí ,j » Lṕ ,q . We again define an equivalence relation on the set t1, . . . , du by i " j if and only if k # i " k # j . We can parameterize the classes of simple modules using the equivalence classes of ": ‚ The c-cellular character corresponding to the class L O is where O 1 is an equivalence class for " such that k # i´k # j " c 0 (if such a class exists, it is unique).
The c-cellular character corresponding to the class L 1 O is where O 1 is an equivalence class for " such that k # i´k # j "´c 0 (if such a class exists, it is unique).
If c 2 0 ‰ pk # i´k # j q 2 , the c-cellular character corresponding to the class L O,O 1 is By definition, every c-cellular character is equal to one of those above.

3.2.
Vectors of height 2 of the canonical basis. Now, we turn to the Leclerc-Miyachi constructible characters for Gpd, 1, 2q and we use the notation of Section 2. Fix r " pr 1 , r 2 , . . . , r d q P Z d and we compute the vectors of the canonical basis of height 2 of V pΛ r q.
We set some notations for the d-symbols of height 2. Let 0 " i 0 ă i 1 ă¨¨¨ă i p " d such that for all 1 ď k ď p´1 we have r i k`1 ă r i k and for all 1 ď k ď p and i k´1 ă i ď i k we have r i " r i k . By convention, we let r 0 "´8 and r d`1 "`8. The d-symbols S " pβ i q 1ďiďd of height 2 are the following: ‚ for 1 ď i ă j ď d, the d-symbol S i,j with β i,r i " r i`1 , β j,r j " r j`1 and β k,l " l for all other values of k and l, ‚ for 1 ď i ď d, the d-symbol S i with β i,r i " r i`2 and β k,l " l for all other values of k and l, ‚ for 1 ď i ď d, the d-symbol S 1 i with β i,r i " r i`1 , β i,r i´1 " r i and β k,l " l for all other values of k and l.
Among these symbols, the following are standard: ‚ for 1 ď k ă l ď p the d-symbol S i k ,i l is standard, ‚ for 1 ď k ď p such that i k´ik´1 ě 2 the d-symbol S i k´1 ,i k is standard, ‚ for 1 ď k ď p the d-symbol S i k is standard, ‚ for 1 ď k ď p such that r i k´r i k`1 ě 2 the d-symbol S 1 i k is standard. For these standard d-symbols, we now apply the algorithm of [11] and show that the element A Σ of the intermediate basis already satisfies A Σ " v Σ mod qF R pΛ r q.
We denote byS i the symbol of height 1 with β i,r i " i`1 and β k,l " l for all other values of k, l. If 1 ď k ď p, we have Let 1 ď k ă l ď p and consider Σ " From the above formula and (H 1 ), one obtains and it is readily checked that A Σ " v Σ mod qF R pΛ r q. Note that if r i k " r i l`1 then k " l`1. Let 1 ď k ď p such that i k´ik´1 ě 2 and consider Σ " S i k´1 ,i k . We obtain pq`q´1qA Σ " so that from the above formula and (H 1 ), one obtains It is readily checked that A Σ " v Σ mod qF R pΛ r q. Let 1 ď k ď p and consider Σ " and we indeed have A Σ " v Σ mod qF R pΛ r q. Finally, let 1 ď k ď p such that r i k´r i k`1 ě 2 and consider Σ " S 1 i k . We obtain From this, we obtain the Leclerc-Miyachi r-constructible characters. The bijections between d-symbols of height 2 and irreducible characters of Gpd, 1, 2q is given by S i,j Ø χ i,j , S i Ø χ i and S 1 i Ø χ 1 i . Proposition 3.6. For 1 ď k ă l ď p, the Leclerc-Miyachi r-constructible character corresponding to the standard d-symbol S i k ,i l is For 1 ď k ď p such that i k´ik´1 ě 2, the Leclerc-Miyachi r-constructible character corresponding to the standard d-symbol S i k´1 ,i k is For 1 ď k ď p, the Leclerc-Miyachi r-constructible character corresponding to the standard d-symbol S i k is For 1 ď k ď p such that r i k´r i k`1 ě 2, the Leclerc-Miyachi r-constructible character corresponding to the standard d-symbol S 1 i k is

A conjecture relating cellular and constructible characters
We now state precisely the conjecture relating Calogero-Moser c-cellular characters and Leclerc-Miyachi r-constructible characters for the complex reflection group Gpd, 1, nq. Let c : RefpGpd, 1, nqq Ñ C. We suppose that c 0 ‰ 0 and that for every 1 ď i ď d we have k i P´Nc 0 . Finally, suppose also that Conjecture 4.1. Let r "´c´1 0 pk # 1 , k # 2 , . . . , k # d q. Then the set of Calogero-Moser c-cellular characters and the set of Leclerc-Miyachi r-constructible characters coincide.
If d " 2 this conjecture is equivalent to the conjecture that Calogero-Moser c-cellular characters for the Weyl group of type B n are Lusztig's constructible characters obtained via truncated induction [13,Chapter 22].
Remark 4.2. If we start from a d-tuple r " pr 1 , . . . , r d q, one can choose a corresponding parameter c by c 0 ‰ 0 and k # i "´c 0 r i . Theorem 4.3. If the parameter c is generic in the sense of Corollary 1.11 then Conjecture 4.1 is true.
For the group Gpd, 1, 2q the Conjecture 4.1 is true for any c.
Proof. With the change of parameters between c and r, the generic case for the c-cellular characters translates into the asymptotic case for the constructible characters. The result therefore follows from Corollary 1.11 and Corollary 2.4. For Gpd, 1, 2q, we describe the equivalence relation " introduced in Section 3.1.2. Using the notation pi j q´1 ďjďp`1 introduced in Section 3.2, the equivalence classes of " are the sets O j " ti j´1`1 , i j´1`2 , . . . , i j u for 1 ď j ď p.
Using the explicit descriptions of the c-cellular characters given in Proposition 3.5 and of the r-constructible characters given in Proposition 3.6 we check that: ‚ for any 1 ď k ă l ď p, the Leclerc-Miyachi r-constructible character γ S i k ,i l is equal to the Calogero-Moser c-cellular character γ 1 O k if r i k " r i l`1 and to the Calogero-Moser c-cellular character γ O k ,O l otherwise, ‚ for any 1 ď k ď p such that i k´ik´1 ě 2, the Leclerc-Miyachi r-constructible character γ S i k´1 ,i k is equal to the Calogero-Moser c-cellular character γ O k ,O k , ‚ for any 1 ď k ă l ď p, the Leclerc-Miyachi r-constructible character γ S i k is equal to the Calogero-Moser c-cellular character γ O k , ‚ for any 1 ď k ă l ď p such that i k´ik´1 ě 2, the Leclerc-Miyachi r-constructible character γ S 1 i k is equal to the Calogero-Moser c-cellular character γ 1 O k . It is easy to check that every Calogero-Moser c-cellular character appears as a Leclerc-Miyachi r-constructible character.

Appendix A. Cellular characters
This appendix aims to define a general notion of cellular characters of a commutative algebra A, and is largely inspired from [5, Appendix II]. We fix k a field of characteristic 0, E a finite dimensional k-vector space, A a split subalgebra of End k pEq and P an integral and integrally closed subalgebra with fraction field K.
Given R a commutative k-algebra, we denote by RE (resp. RA) the extension of scalars R b k E (resp. R b k A). Let D 1 , . . . , D n be some pairwise commuting elements of End P A pP Eq. If p is a prime ideal of P , we denote by K p ppq the residue field at p and by D i ppq the image of D i in End P {pA pP {pEq. Finally, let D " pD 1 , . . . , D n q and P rDs be the subalgebra of End P A pP Eq generated by D 1 , . . . , D n .
We are interested in the decomposition of the vector space KE as a KrDs b K KAmodule, and more precisely of its class in the Grothendieck group K 0 pKrDs b K KAq of finite dimensional KrDs b K KA-modules.
Since the algebra A is split, we obtain (cf. [7,Propositions 3.56 and 7.7]) a bijection IrrpKrDsqˆIrrpKAq Ñ IrrpKrDs b K KAq given by tensoring modules. This bijection induces an isomorphism of Z-modules K 0 pKrDsq b Z K 0 pKAq Ñ K 0 pKrDs b K KAq. We therefore decompose rKEs in K 0 pKrDsq b Z K 0 pKAq as follows: rKEs " ÿ LPIrrpKrDsq rLs b γ P rDs L , with γ P rDs L P K 0 pKAq. Since A is split, we usually think of γ P rDs L as an element of K 0 pAq.
Definition A.1. The set of cellular characters for P rDs is the set of γ P rDs L P K 0 pAq for L running over the set of irreducible KrDs-modules.
Remark A.2. It may happen that γ P rDs L " γ P rDs L 1 for two non-isomorphic KrDs-modules.
By extending the scalars to P rXs " P rX 1 , . . . , X n s and by setting D " X 1 D 1`¨¨¨X n D n , it is shown in [5] that the set of cellular characters for P rDs coincides with the set of cellular characters for P rXsr Ds. We therefore may and will suppose that n " 1 and set D " D 1 .
It is now easy to describe all the irreducible KrDs-modules. Denote by Π the characteristic polynomial of D, which is a unital polynomial in P rts. We then decompose Π into a product of irreducible unital polynomials in Krts Π " Π n 1 1¨¨¨Π nr r . We also denote by Π sem the product of the Π i 's without multiplicity. Since P is integrally closed, the polynomials Π i and Π sem have their coefficients in P and we set L i " P rts{xΠ i y. The set of irreducible KrDs-modules are therefore the extensions to K of the P rDs-modules L i : IrrpKrDsq " tKL 1 , . . . , KL r u.
The main advantage of working over P is that we can easily reduce modulo a prime ideal p of P . Let ∆ be the discriminant of the polynomial Π sem , and denote by ∆ppq its reduction modulo a prime ideal p of P .
Proposition A.3. Let p be a prime ideal of P such that P {p is integrally closed. Then the cellular characters for P {prDppqs are sums of cellular characters for P rDs. If moreover ∆ppq ‰ 0 then the sets of cellular characters for P {prDppqs and for P rDs coincide.
Proof. We start by decomposing the reduction Πppq modulo p into a product of irreducible polynomials with coefficients in k P ppq: where π i,j P k P ppqrts is unital and irreducible, e i,j P Z ą0 and π i,j ‰ π i,j 1 for j ‰ j 1 . Since P {p is integrally closed, the polynomials π i,j have their coefficients in P {p.
For 1 ď i ď r and 1 ď j ď d i , we denote by L i,j the P {prDppqs-module pP {pqrts{xπ i,j y. In the Grothendieck group of K P ppqrDppqs we therefore have the following equality (2) rk P ppqL i s " d i ÿ j"1 e i,j rk P ppqL i,j s.
Since k has characteristic 0, an equality between elements of K 0 pKrDsb K KAq is equivalent to an equality between the corresponding characters, so that we can specialize modulo p the equality rKEs " Using (2), we see that the cellular characters for P {prDppqs are sums of cellular characters for P rDs.
If moreover ∆ppq ‰ 0, then e i,j " 1 for all 1 ď i ď r and 1 ď j ď d i and π i,j " π l,m if and only if pi, jq " pl, mq. Then rk P ppqEs " r ÿ i"1 d i ÿ j"1 rk P ppqL i,j s b γ P rDs i and the set tk P ppqL i,j | 1 ď i ď r, 1 ď j ď d i u is exactly the set of irreducible representations of k P ppqrDppqs, so that γ P {prDppqs k P ppqL i,j " γ P rDs KL i for all 1 ď i ď r and 1 ď j ď d i .