Canonical bases of higher-level q -deformed Fock spaces

DOI 10.1007/s10801-007-0062-7

Canonical bases of higher-level q -deformed Fock spaces

Xavier Yvonne

Received: 12 June 2006 / Accepted: 29 January 2007 / Published online: 17 April 2007

© Springer Science+Business Media, LLC 2007

Abstract We show that the transition matrices between the standard and the canon- ical bases of infinitely many weight subspaces of the higher-levelq-deformed Fock spaces are equal.

1 Introduction

The q-deformed higher-level Fock spaces were introduced in [6] in order to com- pute the crystal graph of any irreducible integrable representation of level l ≥1 of Uq(sln). More precisely, the Fock representation Fq[s_l]depends on a parameter s_l=(s₁, . . . , s_l)∈Z^l called multi-charge. It contains as a submodule the irreducible integrableU_q(sl_n)-module with highest weight_s1+ · · · +_sl. The representation Fq[s_l]is a generalization of the level-one Fock representation ofU_q(sl_n)([4,17], see also [14,15]).

The canonical bases are bases of the Fock representations that are invariant under a certain involution ofU_q(sl_n)and that give atq=0 andq= ∞the crystal bases.

They were constructed forl=1 in [14,15] and forl≥1 by Uglov [19]. In [19], Uglov provides an algorithm for computing these canonical bases. He also gives an expression of the transition matrices between the standard and the canonical bases in terms of Kazhdan-Lusztig polynomials for affine Hecke algebras of typeA.

In this article we prove three theorems.

1. The first one (Theorem 3.9) is a generalization to l≥1 of a result of [13]. It compares the transition matrices of the canonical bases of some weight subspaces inside a given Fock space Fq[s_l]. The weights involved are conjugated under the action of the Weyl group ofU_q(sl_n). This action leads to bijectionsσ_i that can

X. Yvonne (

Institut Camille Jordan (Mathématiques), Université Lyon I, 43 Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France

e-mail: yvonne@math.univ-lyon1.fr

be described in a combinatorial way by adding/removing as manyi-nodes as pos- sible to thel-multi-partitions indexing the canonical bases. These bijections are generalizations of the Scopes bijections introduced in [18] in order to study, when n=pis a prime number, thep-blocks of symmetric groups of a given defect.

2. In a dual manner, our second result (Theorem 4.4) gives some sufficient condi- tions on multi-chargess_landt_lwith given residues modulonthat ensure that the transition matrices of the canonical bases of some weight subspaces of Fq[sl]and Fq[t_l]coincide.

3. Our third result (Theorem5.2) is an application of Theorem4.4to the case when the multi-chargess_l=(s₁, . . . , s_l)andt_l=(t₁, . . . , t_l)are dominant, that iss₁

· · · s_l andt₁ · · · t_l. It shows that the transition matrices of the canonical bases of the Fock spaces Fq[s_l]stabilize whens_lbecomes dominant (with a given sequence of residues modulon). This supports the following conjecture (see [22]).

We conjecture that ifs_l=(s1, . . . , sl)is dominant, then the transition matrix of the homogeneous component of degreemof the canonical basis of the Fock space Fq[s_l] is equal to the decomposition matrix of the cyclotomic v-Schur algebra SC,m(ζ;ζ^s1, . . . , ζ^sl)of [2], whereζis a complex primitiven-th root of unity. This conjecture generalizes both Ariki’s theorem for Ariki-Koike algebras (see [1]) and a result of Varagnolo and Vasserot (see [20]) which relates the canonical basis of the level-one Fock space and the decomposition matrix ofv-Schur algebras with parameter a complexn-th root of unity.

Notation LetN(respectivelyN^∗) denote the set of nonnegative (respectively posi- tive) integers, and fora, b∈Rdenote by[[a;b]]the discrete interval[a, b] ∩Z. For X⊂R,t∈R,N∈N^∗, put

X^N(t ):= {(s1, . . . , sN)∈X^N|s1+ · · · +sN=t}. (1) Throughout this article, we fix 3 integersn,l≥1 ands∈Z. Letdenote the set of all integer partitions, and forN∈N^∗, let^Ndenote the set of allN-multi-partitions.

The empty partition (respectively emptyN-multi-partition) will be denoted by∅(re- spectively∅N).

2 Higher-levelq-deformed Fock spaces

In this section, we introduce the higher-level Fock spaces and their canonical bases.

We follow here [19], to which we refer the reader for more details. All definitions and results given here are due to Uglov.

2.1 The quantum algebrasU_q(sl_n)andU_p(sl_l)

In this section, we assume thatn≥2 andl≥2. Letslnbe the Kac-Moody algebra of typeA⁽¹⁾_n−1defined over the fieldQ[7]. Leth^∗be the dual of the Cartan subalgebra ofsl_n. Let₀, . . . , _n−1∈h^∗be the fundamental weights,α₀, . . . , α_n−1∈h^∗be the simple roots andδ:=α₀+ · · · +α_n−1be the null root. It will be convenient to extend

the index set of the fundamental weights by setting_i:=_imodnfor alli∈Z. The simple roots are related to the fundamental weights by

αi=2i−i−1−i+1+δi,0δ (0≤i≤n−1). (2) For 0≤i, j≤n−1, letai,j be the coefficient ofj inαi. The space

h^∗=

n−1

i=0

Qi ⊕ Qδ=

n−1

i=0

Qαi ⊕ Q0

is equipped with a non-degenerate bilinear symmetric form(. , .)defined by (αi, αj)=ai,j, (0, αi)=δi,0, (0, 0)=0 (0≤i, j≤n−1). (3) Let U_q(sl_n)be the q-deformed universal enveloping algebra of sl_n. This is an algebra over Q(q)with generatorsei,fi,t_i^±1 (0≤i≤n−1)and∂. Let U_q(sln) be the subalgebra ofU_q(sln)generated bye_i,f_i,t_i^±1(0≤i≤n−1). The relations inU_q(sl_n)are standard and will be omitted (see e.g. [10]). The relations among the degree generator∂ and the generators ofU_q(sl_n)can be found in [19, §2.1]. IfMis aU_q(sln)-module, denote byP(M)the set of weights ofMand letMwdenote the subspace ofMof weightw. Ifx∈Mw \ {0}is a weight vector, denote by

wt(x):=w (4)

the weight ofx. The Weyl group ofsln(orUq(sln)), denoted byWn, is the subgroup of GL(h^∗)generated by the simple reflectionsσi defined by

σ_i()=−(, α_i) α_i (∈h^∗,0≤i≤n−1). (5) Note thatW_n is isomorphic toS_n, the affine symmetric group which is a Coxeter group of typeA⁽¹⁾_n−1.

We also introduce the algebraUp(sll)with

p:= −q⁻¹. (6)

In order to distinguish the elements related toU_q(sl_n)from those related toU_p(sl_l), we put dots over the latter. For example, e˙_i,f˙_i,t˙_i^±1 (0≤i≤l−1)and∂˙ are the generators of U_p(sl_l), α˙_i (0≤i≤l−1) are the simple roots for U_p(sl_l), W˙_l = ˙σ0, . . . ,σ˙_l−1is the Weyl group ofU_p(sl_l)and so on. Similarly, ifM is aU_p(sl_l)- module, denote byP˙(M)the set of weights ofM.

2.2 The space^s

2.2.1 The vector space^sand its standard basis

Following [19], we now recall the definition of^s, the space of semi-infiniteq-wedge products of charges(this space is denoted by^s+∞2 in [19]). First, letr≥2 be an

integer, and^r_qV be the space ofq-wedge products of finite lengthr (this space is denoted by^r in [19]; we hope that this does not make any confusion with our^s).

As a vector space overQ(q),^r_qV is spanned by theq-wedge products

u_k=u_k1∧u_k2∧ · · · ∧u_kr, k=(k₁, k₂, . . . , k_r)∈Z^r, (7) with relations given in [19, Prop. 3.16]. These relations are called straightening rules (we will not need them in this article). Now, define^sas the inductive limit

^s=lim

−→^r_qV , (8)

where maps ^r_qV →^r_qV (r> r) are given by v→v∧u_s−r ∧u_s−r−1∧ · · · ∧ u_s−r+1. Less formally,^s is spanned byq-wedge products of infinite length

u_k=u_k1∧u_k2∧ · · ·, k=(k₁, k₂, . . .)∈P (s), (9) whereP (s)is the set of all sequences of integers(k1, k2, . . .)such thatki=s−i+1 forilarge enough. The straightening rules given in [19, Prop. 3.16 (i)] still hold for any pair of adjacent factors of aq-wedge productu_k∈^s. From now on, we shall assume without further comment that allq-wedge products lie in^s (in particular, they have infinitely many factors). Using the straightening rules, one can express aq-wedge product as a linear combination of so-called ordered q-wedge products, namelyq-wedge productsukwithk∈P⁺⁺(s), where

P⁺⁺(s):= {(k1, k2, . . .)∈P (s)|k1> k2>· · ·}. (10) In fact, the orderedq-wedge products{u_k|k∈P⁺⁺(s)}form a basis of^s, called the standard basis. In this article, it will be convenient to use different indexations of this basis which we give now.

* Indexationu_k. This is the indexation we have just described.

* Indexationλ. To the ordered q-wedge product u_k corresponds a partition λ= (λ1, λ2, . . .)defined by

λ_i:=k_i−(s+1−i) (i≥1). (11) Ifu_kandλare related this way, write

|λ, s :=uk. (12)

* Indexationλn. Recall the definition ofZⁿ(s)from (1). Uglov constructed a bijec- tion

τ_n :→ⁿ×Zⁿ(s), λ→(λ_n,s_n) (13) (see [19, §4.1], where this map is denoted byτ_n^s). With the notation above,λ_nis the n-quotient ofλands_nis a variation of then-core ofλ(see e.g. [16, Ex.8, p.12]).

Write

|λn,sn^•:= |λ, s (14)

if (λ_n,s_n)=τ_n(λ). Note that this indexation coincides with the indexationλ if n=1.

* Indexationλ_l. Uglov constructed a bijection

τ_l:→^l×Z^l(s), λ→(λ_l,s_l) (15) (see again [19, §4.1], where this map is denoted byτ_l^s). The mapτ_l is a variation of the mapτ_n defined above. Write

|λ_l,s_l := |λ, s (16) if(λ_l,s_l)=τ_l(λ). Note that this indexation coincides with the indexationλifl=1.

Example 2.1 Taken=2,l=3 ands= −1. Then we have

u₃∧u₁∧u₀∧u₋₂∧u₋₄∧u₋₆∧u₋₇∧ · · · =(4,3,3,2,1),−1

=(3,3),∅

, (−1,0)_•

=(1,1), (1,1), (1)

, (0,0,−1) .

2.2.2 Three actions on^s

Following [3, 6, 19], the vector space ^s can be made into an integrable repre- sentation of level l of the quantum algebra Uq(sln). This representation can be described in a nice way if we use the indexation λ_l. In order to recall the ex- plicit formulas, let us first introduce some notation. Fix λ_l =(λ⁽¹⁾, . . . , λ^(l))∈^l and sl =(s1, . . . , sl)∈Z^l. Identify the multi-partitionλl with its Young diagram {(i, j, b)∈N^∗×N^∗× [[1;l]] |1≤j ≤λ^(b)_i }, whose elements are called nodes of λ_l. For each nodeγ=(i, j, b)ofλ_l, define its residue modulonby

resn(γ ) = resn(γ ,s_l) := (sb+j−i) modn ∈ Z/nZ∼= [[0;n−1]]. (17) If resn(γ )=c, we say thatγ is ac-node. Ifμ_l∈^l is such thatμ_l⊃λ_l andγ :=

μ_l\λ_l is ac-node ofμ_l, we say thatγ is a removablec-node ofμ_l or thatγ is an addablec-node ofλ_l. For 0≤c≤n−1, denote by

M_c(λ_l;s_l;n) (respectivelyA_c(λ_l;s_l;n), respectivelyR_c(λ_l;s_l;n)) (18) the number ofc-nodes (respectively of addablec-nodes, respectively of removable c-nodes) ofλ_l. Put

Nc(λ_l;s_l;n):=Ac(λ_l;s_l;n)−Rc(λ_l;s_l;n). (19) Forλ_l,μ_l∈^l,s_l∈Z^l,c∈ [[0;n−1]]andk∈N^∗, write

λ_l−→^c:k μ_l (20)

if there exists a sequence ofl-multi-partitionsν⁽⁰⁾_l ⊂ν⁽¹⁾_l ⊂ · · · ⊂ν^(k)_l such thatλ_l= ν⁽⁰⁾_l ,μ_l =ν^(k)_l and for all 1≤j ≤k,ν^{(j )}_l \ν^(j_l ⁻¹⁾ is an addablec-node of ν^(j_l ⁻¹⁾.

Given a multi-charge(s₁, . . . , s_l)and two nodes γ =(i, j, b) andγ=(i, j, b), write

γ < γ (21)

if eithers_b+j −i < s_b +j−i or s_b+j−i=s_b +j−i andb < b. This defines a total ordering on the set of the addable and removablec-nodes of a given multi-partition. Ifλ_l−→^c:k μ_l, put

N_c^>(λ_l;μ_l;s_l;n)=

γ∈μ_l\λ_l

{β∈N³|βis an addablec-node ofμ_landβ > γ}

−{β∈N³|β is a removablec-node ofλlandβ > γ} ,(22)

and define similarlyN_c^<(λ_l;μ_l;s_l;n).

Example 2.2 Take s_l =(5,0,2,1), λ_l =

(5,3,3,1), (3,2), (4,3,1), (2,2,2,1) , n=3 andc=0. Then we have

M_c(λ_l;s_l;n)=11, A_c(λ_l;s_l;n)=R_c(λ_l;s_l;n)=5 and N_c(λ_l;s_l;n)=0.

The addablec-nodes ofλ_lare(5,1,4),(4,2,1),(1,4,2),(1,3,4)and(1,5,3). The removablec-nodes ofλ_l are (2,2,2),(3,1,3),(3,2,4),(2,3,3)and(1,5,1). The list of all these nodes arranged with respect to the ordering described above is

(5,1,4) < (2,2,2) < (3,1,3) < (3,2,4) < (4,2,1) < (1,4,2)

< (2,3,3) < (1,3,4) < (1,5,3) < (1,5,1).

Take alsoμ_l=

(5,3,3,1), (3,2), (5,3,1), (2,2,2,1)

, so thatμ_l\λ_l= {(1,5,3)} is a single c-node. Then N_c^>(λ_l;μ_l;s_l;n)=0−1= −1 and N_c^<(λ_l;μ_l;s_l;n)=

4−4=0.

Fors_l=(s₁, . . . , s_l)∈Z^l, define (s_l, n):=1

2

l

b=1

s_b² n −sb

−(s_bmodn)²

n −(sbmodn)

. (23)

Now we can state the following result.

Theorem 2.3 [3,6,19] The following formulas define on^s a structure of an inte- grable representation of levellof the quantum algebraU_q(sl_n).

e_i.|ν_l,s_l =

λ_l−→^i:1ν_l

q^{−Ni<(λl;νl;sl;n)}|λ_l,s_l, (24)

f_i.|ν_l,s_l =

ν_l−→^i:1 μ_l

q^{Ni>(νl;μl;sl;n)}|μ_l,s_l, (25)

t_i.|ν_l,s_l =q^Ni(νl;sl;n)|ν_l,s_l, (26)

∂.|ν_l,s_l = −

(s_l, n)+M₀(ν_l;s_l;n)

|ν_l,s_l. (27) Note that these formulas involve no straightening of q-wedge products. They are therefore handy to use for computations.

In a completely similar way, ^s can be made into an integrable representa- tion of level n of the quantum algebraU_p(sll). This action can be described us- ing the indexation λ_n. Namely, we have (with obvious notation) the following re- sult.

Theorem 2.4 [3,6,19] The following formulas define on^s a structure of an inte- grable representation of levelnof the quantum algebraUp(sll).

˙

e_i.|ν_n,s_n^•=

λn i:1

−→νn

p^{−Ni<(λn;νn;sn;l)}|λ_n,s_n^•, (28)

f˙_i.|ν_n,s_n^•=

νn i:1

−→μ_n

p^{Ni>(νn;μn;sn;l)}|μ_n,s_n^•, (29)

˙

t_i.|ν_n,s_n^•=p^Ni(νn;sn;l)|ν_n,s_n^•, (30)

∂.˙|ν_n,s_n^•= −

(s_n, l)+M0(ν_n;s_n;l)

|ν_n,s_n^•. (31) Theorems2.3and2.4show in particular that the vectors of the standard basis of ^s are weight vectors for the actions of Uq(sln)andUp(sll), and the weights are given by:

Corollary 2.5 [19], (27–30) With obvious notation, we have

wt(|λ_l,s_l)= −(s_l, n)δ+_s1+ · · · +_sl−

n−1

i=0

M_i(λ_l;s_l;n) α_i, (32)

˙

wt(|λ_l,s_l)= −

(s_l, n)+M₀(λ_l;s_l;n)δ˙+(n−s₁+s_l)˙₀ +

l−1

i=1

(s_i−s_i+1)˙_i, (33)

˙

wt(|λ_n,s_n^•)= −(s_n, l)˙δ+ ˙_s1+ · · · + ˙_sn−

l−1

i=0

M_i(λ_n;s_n;l)α˙_i, (34)

wt(|λ_n,s_n^•)= −

(s_n, l)+M₀(λ_n;s_n;l)

δ+(l−s₁+s_n)₀ +

n−1

i=1

(s_i−s_i+1) _i. (35)

Definition 2.6 Form∈Z^∗, define an endomorphismB_mof^sby

Bm(uk1∧uk2∧ · · ·):=^+∞

j=1

uk1∧ · · · ∧uk_j−1∧uk_j−nlm∧uk_j+1∧ · · · (k₁, k₂, . . .)∈P⁺⁺(s)

. (36)

Using a variation of [19, Lemma 3.18] forq-wedge products with infinitely many fac- tors, one sees that the sum above involves only finitely many nonzero terms, hence B_m is well-defined. This definition comes from a passage to the limit r→ ∞ in the action of the center of the Hecke algebra ofS_r onq-wedge products ofr fac- tors. However, the operatorsB_mdo not commute, but by [19, Prop. 4.4], they span a Heisenberg algebra

H:= B_m|m∈Z^∗. (37) We now recall some results concerning the actions of U_q(sl_n),U_p(sl_l)and H on^s.

Proposition 2.7 [19], Prop. 4.6 Recall thatp= −q⁻¹. Then the actions ofU_q(sl_n),

U_p(sl_l)andHon^s pairwise commute.

ForL,N∈N^∗, introduce the finite set

A_L,N(s):= {(r₁, . . . , r_L)∈Z^L(s)|r₁≥ · · · ≥r_L, r₁−r_L≤N}. (38) Using [19, §4.1], it is not hard to see that ifr_l∈Z^l(s)andr_n∈Zⁿ(s)are such that

|∅l,r_l = |∅n,r_n^•, thenr_l∈A_l,n(s)andr_n∈A_n,l(s). Conversely, ifr_l∈A_l,n(s), then there exists a unique r_n∈A_n,l(s) such that |∅n,r_n^•= |∅l,r_l, and if r_n∈ A_n,l(s), then there exists a uniquer_l∈A_l,n(s)such that|∅l,r_l = |∅n,r_n^•. There- fore,

{|∅l,r_l |r_l∈A_l,n(s)} = {|∅n,r_n^•|r_n∈A_n,l(s)}

is a set of highest weight vectors simultaneously for the actions of U_q(sl_n) and U_p(sl_l). It is easy to see that these vectors are also singular for the action of H, that is, they are annihilated by theB_m,m >0. It turns out that these vectors are the only singular vectors simultaneously for the actions ofU_q(sl_n),U_p(sl_l)andH, and we have the following theorem.

Theorem 2.8 [19], Thm. 4.8 We have

^s=

r_l∈A_l,n(s)

U_q(sl_n)⊗H⊗U_p(sl_l).|∅l,r_l

=

rn∈An,l(s)

U_q(sln)⊗H⊗U_p(sll).|∅n,r_n^•.

2.2.3 The involution of^s

Following [19], the space^s can be endowed with an involution . Instead of recall- ing the definition of this involution, we give its main properties (by [23, Thm. 3.11], they turn out to characterize it completely).

Proposition 2.9 [19] There exists an involution of^s such that:

(i) is aQ-linear map of^ssuch that for allu∈^s,k∈Z, we haveq^ku=q^−ku.

(ii) (Unitriangularity property). For allλ∈, we have

|λ, s ∈ |λ, s +

μλ

Z[q, q⁻¹] |μ, s, wherestands for the dominance ordering on partitions.

(iii) For allλ∈, we have wt(|λ, s)=wt(|λ, s)andwt(˙ |λ, s)= ˙wt(|λ, s).

(iv) For all 0≤i≤n−1, 0≤j≤l−1,m <0,v∈^s, we have f_i.v=f_i.v, f˙_j.v= ˙f_j.v and B_m.v=B_m.v.

Proof Let be the involution of^s defined in [19, Prop. 3.23 & Eqn. (39)]. By construction, (i) holds. The other statements come from [19, Prop. 4.11 & 4.12] and

Corollary2.5.

2.3 q-deformed higher-level Fock spaces 2.3.1 Definition

By Theorem2.3, the space Fq[s_l] :=

λ_l∈^l

Q(q)|λ_l,s_l ⊂^s (s_l∈Z^l(s)) (39)

is a U_q(sln)-submodule of ^s. The reader should be aware that Fq[s_l] is not a U_p(sl_l)-submodule of^s. In a similar way, by Theorem2.4, the space

Fp[s_n]^•:=

λn∈ⁿ

Q(q)|λ_n,s_n^•⊂^s (s_n∈Zⁿ(s)) (40)

is aU_p(sl_l)-submodule of^s.

Definition 2.10 [19] The representations Fq[s_l]and Fp[s_n]^•(sl∈Z^l(s),s_n∈Zⁿ(s)) are called (q-deformed) Fock spaces. Whenl >1 andn >1, we speak of higher-level

Fock spaces.

Since the mapsτlandτ_n are bijections, we have the following decompositions:

^s=

s_l∈Z^l(s)

Fq[s_l] =

s_n∈Zⁿ(s)

Fp[s_n]^•. (41) Neither of these decompositions is compatible with the decompositions of^s given in Theorem2.8.

2.3.2 Fock spaces as weight subspaces of^s. Actions of the Weyl groups.

LetN,L∈N^∗. Recall the definition ofQ^L(s)andQ^L(N )from (1) and define a map θ_L,N :Q^L(s)→Q^L(N ), (s1, . . . , s_L)→(N−s1+s_L, s1−s2, . . . , s_L−1−s_L).

(42) (Note that this map also depends on the charge s∈Z that we have fixed. How- ever, since s will not vary in this paper, we do not keep it in our notation.) It is easy to see thatθ_L,N is bijective. Moreover, for(a₁, . . . , a_L)∈Q^L(N ), thel-tuple (s1, . . . , s_L):=θ_L,N⁻¹(a1, . . . , a_L)is given by

s_i= 1 L

⎛

⎝s−

L−1

j=1

j a_j+1

⎞

⎠+

L

j=i+1

a_j (1≤i≤L). (43) The next result shows that the Fock spaces are sums of certain weight subspaces of ^s. The proof follows easily from Corollary2.5.

Proposition 2.11 [19]

(i) Lets_n∈Zⁿ(s). Let(a₁, . . . , a_n):=θ_n,l(s_n)andw:=_n

i=1a_ii−1. Then Fp[s_n]^•=

d∈Z

^sw+dδ. (ii) Lets_l∈Z^l(s). Let(a1, . . . , al):=θl,n(s_l)andw˙:=_l

i=1ai˙i−1. Then Fq[s_l] =

d∈Z

^s ˙w+dδ˙.

Note that the operator B_m (m∈Z^∗)maps the weight subspace ^sw (respec- tively ^s ˙w) into^sw+mδ (respectively^s ˙w+m˙δ). Therefore, by Propo- sition2.11, the Fock spaces Fq[s_l]and Fp[s_n]^• (sl∈Z^l(s),s_n∈Zⁿ(s)) are stable under the action ofH.

We now compare some weight subspaces of the Fock spaces. The proof follows again from Corollary2.5.

Proposition 2.12

(i) Let s_l =(s₁, . . . , s_l)∈Z^l(s)and w be a weight of Fq[s_l]. Then there exists a unique pair(s_n,w)˙ such that Fq[s_l]w =Fp[s_n]^• ˙w, wheres_nis inZⁿ(s)and

˙

w is a weight of Fp[s_n]^•. More precisely, write w=dδ+_n

i=1a_ii−1 with a1, . . . , an, d∈Z, and puts0:=n+sl. Then we haves_n=θ_n,l⁻¹(a1, . . . , an)and

˙

w=dδ˙+_l−1

i=0(s_i−s_i+1)˙_i.

(ii) Lets_n=(s₁, . . . , s_n)∈Zⁿ(s)andw˙ be a weight of Fp[s_n]^•. Then there exists a unique pair(s_l, w) such that Fp[s_n]^• ˙w =Fq[s_l]w, where s_l is in Z^l(s) andwis a weight of Fq[s_l]. More precisely, writew˙ =dδ˙+_l

i=1ai˙i−1with a1, . . . , a_l, d ∈Z, and puts0:=l+s_n. Then we haves_l=θ_l,n⁻¹(a1, . . . , a_l)and w=dδ+_n−1

i=0(s_i−s_i+1)_i.

Example 2.13 Take n=3, l=2, s_l =(1,0) andw= −20+1+32−2δ. Then by (32), we have wt(1,1), (1)

,s_l

=w, sow is a weight of Fq[s_l]. By Proposition 2.12 (i), we have Fq[s_l]w =Fp[s_n]^• ˙w with s_n=(2,1,−2) and

˙

w=2˙₀+ ˙₁−2δ. Moreover, using (32) and (34), we see that for all˙ |λ_l,s_l =

|λ_n,s_n^•∈Fq[s_l]w =Fp[s_n]^• ˙w, we haveM₀(λ_l;s_l;n)=2, M₁(λ_l;s_l;n)=1, M₂(λ_l;s_l;n)=0 andM₀(λ_n;s_n;l)=M₁(λ_n;s_n;l)=0 (this shows a posteriori that

dim(Fq[s_l]w)=1).

We now deal with the actions of the Weyl groups ofU_q(sl_n)andU_p(sl_l)on the set of the weight subspaces of^s. Recall thatW_n= σ0, . . . , σ_n−1is the Weyl group of U_q(sl_n). Since theα_i’s are the simple roots and the_j’s are the fundamental weights for the Kac-Moody algebrasl_n, we have (_j, α_i)=δ_i,j for all 0≤i, j ≤n−1.

Hence by (5),Wnacts on the weight lattice_n−1

i=0Zi⊕Zδby σ_i.δ=δ and

σ_i._j=

_j ifj=i,

i−1+i+1−i−δi,0δ ifj=i (0≤i, j≤n−1). (44) Moreover, it is easy to see thatW_nacts faithfully onZⁿ(s)by

σ0.(s1, . . . , s_n)=(s_n+l, s2, . . . , s_n−1, s1−l),

σ_i.(s1, . . . , s_n)=(s1, . . . , s_i+1, s_i, . . . , s_n) (1≤i≤n−1), (45) and the set An,l(s)defined by (38) is a fundamental domain for this action. In a similar way, one can define two actions of the Weyl groupW˙lofUp(sll), one on the weight lattice_l−1

i=0Z˙_i⊕Zδ˙and one onZ^l(s). The following lemma will be useful later.

Lemma 2.14 Lets_n∈Zⁿ(s)and w˙ be a weight of Fp[s_n]^•. Let(s_l, w)∈Z^l(s)× P(^s)be the unique pair such that Fp[s_n]^• ˙w =Fq[s_l]w (see Proposition2.12

(i)). Letσ˙ ∈ ˙W_l. In the same way, let(t_l, w)∈Z^l(s)×P(^s)be the unique pair such that Fp[s_n]^• ˙σ .w =˙ Fq[t_l]w. Then we have

t_l= ˙σ .s_l and w=w+wt(|∅l,t_l)−wt(|∅l,s_l).

Proof The proof follows immediately from the formulas given in Proposition2.12.

2.3.3 The lower crystal basis(L[s_l],B[s_l])of Fq[s_l]atq=0

Lets_l∈Z^l(s). Following [8], letA⊂Q(q)be the ring of rational functions which are regular atq=0,L[s_l] :=

λl∈^lA|λ_l,s_land for 0≤i≤n−1, lete˜_i^low,f˜_i^low,

˜

e_i^up andf˜_i^up denote Kashiwara’s operators acting onL[s_l]. The following lemma shows that sometimes, certain powers of the operatorse˜i^low ande˜i^up coincide (one has an analogous result forf˜_i^lowandf˜_i^up).

Lemma 2.15 Let s_l ∈Z^l(s), w∈P(Fq[s_l]), u∈(Kere_i)∩Fq[s_l]w and k :=

(w, α_i). Then we have

(e˜_i^up)^k.(f_i^(k).u)=(e˜_i^low)^k.(f_i^(k).u)=u.

Proof The second equality follows easily by induction onk from the definition of

˜

ei^low. Let us now show that(e˜i^up)^k.(f_i^(k).u)=u. Note that for 0≤j≤k, we have (wt(f_i^{(k−j )}.u), α_i)=(wt(u), αi)−(k−j )(α_i, α_i)=2j−k.

By induction on 0≤k≤k, we get therefore, by definition ofe˜_i^up,

(e˜_i^up)^k.(f_i^(k).u)=

⎛

⎝^k

−1

j=0

[(2j−k)+(k−j )+1]

[k−j]

⎞

⎠f_i^(k−k).u (0≤k≤k).

As a consequence, we have(e˜i^up)^k.(f_i^(k).u)=

⎛

⎝^k−1

j=0

[j+1]

[k−j]

⎞

⎠u=u.

Put

B[s_l] := {|λ_l,s_lmodqL[s_l] |λ_l∈^l}. (46) From now on, we shall write more brieflyλ_lfor the element inB[s_l]indexed by the corresponding multi-partition. By [3,6,19], the pair(L[s_l],B[s_l])is a lower crystal basis of Fq[s_l]atq=0 in the sense of [8], and the crystal graphB[s_l]contains the arrowλ_l−→ⁱ μ_l if and only if the multi-partitionμ_l is obtained fromλ_l by adding a goodi-node in the sense of [19, Thm. 2.4]. We shall still denote bye˜_i^low andf˜_i^low Kashiwara’s operators acting onB[s_l] ∪ {0}.