Stabilization of the Brylinski-Kostant filtration and limit of Lusztig q-analogues

DOI 10.1007/s10801-007-0097-9

Stabilization of the Brylinski-Kostant filtration and limit of Lusztig q-analogues

Cédric Lecouvey

Received: 17 April 2007 / Accepted: 9 August 2007 / Published online: 18 September 2007

© Springer Science+Business Media, LLC 2007

Abstract LetGbe a simple complex classical Lie group with Lie algebragof rankn.

We show that the coefficient of degreekin the Lusztigq-analogueK_λ,μ^g (q)associated to the fixed partitionsλandμstabilizes fornsufficiently large. As a consequence, we obtain the stabilization of the dimensions in the Brylinski-Kostant filtration asso- ciated to any dominant weight. We then introduce, for each pair of partitions(λ, μ), formal series which can be regarded as natural limits of the Lusztigq-analogues. We give a duality property for these limits and recurrence formulas which permit notably to derive explicit expressions whenλis a row or a column partition.

Keywords Brylinski filtration·q-analogues·Lusztig polynomials·q-series 1 Introduction

The multiplicityK_λ,μof the weightμin the irreducible finite-dimensional represen- tationV^g(λ)of the simple Lie groupGwith Lie algebragcan be written in terms of the ordinary Kostant partition functionP defined by the equality

αpositive root

1

(1−e^α)=

β

P(β)e^β

whereβ runs on the set of nonnegative integral combinations of positive roots ofg.

ThusP(β)is the number of ways the weightβcan be expressed as a sum of positive roots. Then, from the Weyl character formula, one derives the identity

K_λ,μ^g =

w∈W^g

(−1)^(w)P(w(λ+ρ)−(μ+ρ)) (1)

C. Lecouvey (

Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville, B.P. 699, 62228 Calais Cedex, France

e-mail: Cedric.Lecouvey@lmpa.univ-littoral.fr

whereW^gis the Weyl group ofg.

The Lusztigq-analogue of weight multiplicityK_λ,μ^g (q)is obtained by substituting the ordinary Kostant partition functionP by itsq-analoguePq^gin (1). NamelyPq^gis defined by the equality

αpositive root

1

(1−qe^α)=

β

Pq^g(β)e^β

and we have

K_λ,μ^g (q)=

w∈W^g

(−1)^(w)Pq^g(w(λ+ρ)−(μ+ρ)). (2)

As shown by Lusztig [14],K_λ,μ^g (q)is a polynomial inqwith nonnegative integer co- efficients. Many interpretations of the Lusztigq-analogues exist. In particular, they can be obtained from the Brylinski-Kostant filtration of weight spaces [1]. The poly- nomialsK_λ,^g_∅(q)appear in the graded character of the harmonic polynomials associ- ated tog[5]. We also recover the Lusztigq-analogues as the coefficients of the ex- pansion of the Hall-Littlewood polynomials on the basis of Weyl characters [7]. This notably permits to prove that they are affine Kazhdan-Lusztig polynomials. In [11]

Lascoux and Schützenberger have obtained a combinatorial expression forK_λ,μ^gln(q) in terms of the charge statistic on the semistandard tableaux of shapeλand evalua- tionμ. By using the combinatorics of crystal graphs introduced by Kashiwara and Nakashima [6], we have also established similar formulas [12,13] for the Lusztig q-analogues associated to the symplectic and orthogonal Lie algebras when(λ, μ) satisfies restrictive constraints.

Considerλ, μtwo partitions of length at mostm. These partitions can be regarded as dominant weights forg=gl_n,so_2n+1,sp_2n orso2nwhen n≥m. Then,K_λ,μ^gln(q) does not depend on the rank n considered. Such a property does not hold for the Lusztigq-analoguesK_λ,μ^g (q)wheng=so_2n+1,sp_2norso2nwhich depend in general on the rank of the Lie algebra considered. Write

K_λ,μ^g (q)=

k≥0

K_λ,μ^g,kq^k.

We first establish in this paper that forg=so_2n+1,sp_2norso2n, the coefficientK_λ,μ^g,k stabilizes whenntends to infinity. More precisely,K_λ,μ^g,k does not depend on the rank n ofgprovidingn≥2k+a wherea is the number of nonzero parts ofμ (Theo- rem4.3.1). Note that this result cannot be obtained by simply taking the limit when ntends to infinity in (2). Indeed, the number of decompositions of a given weight as a sum of kpositive roots may strictly increase withn. Hence, there is no analogue of the Kostant partition function in infinite rank (but see Remark(i)in Section 6.2).

By Brylinski’s interpretation of the coefficients K_λ,μ^g,k [1], one then obtains that the dimension of thek-th component of the Brylinski-Kostant filtration associated to the finite-dimensional irreducible representations ofg=so_2n+1,sp_2n orso_2n stabilizes fornsufficiently large (Theorem4.4.2). Observe that this stabilization is immediate

forg=gl_nsince the polynomialsK_λ,μ^gln(q)do not depend onn. Forg=so_2n+1,sp_2n orso_2nthe Brylinski-Kostant filtration depends in general on the rank considered and it seems difficult to obtain the dimension of its components by direct computations.

Our method is as follows. We obtain the explicit decomposition of the symmetric algebra S(g) considered as a G-module into its irreducible components by using identities due to Littlewood. This permits to show that the multiplicities appearing in the decomposition of thek-th graded componentS^k(g)ofS(g)do not depend on the ranknofgprovidingnis sufficiently large. Observe that this stabilization also fol- lows from a more general result due to Hanlon [3]. Nevertheless our computations, based on Littlewood’s identities, have the interest to yield simple explicit formulas in terms of the Littlewood-Richardson coefficients for the decomposition ofS^k(g)in large rank. Thanks to a classical result by Kostant, we establish a similar result for the k-th graded component H^k(g)of the space H (g)of G-harmonic polynomials.

These stabilization properties are equivalent to the existence of a limit in infinitely many variables for the graded characters associated toS(g)andH (g). The limits so obtained are formal series with coefficients in the ring of universal characters intro- duced by Koike and Terada. From Hesselink expression [5] of the graded charac- ter ofH (g), one then derives thatK_λ,^g,k_∅ stabilizes fornsufficiently large. By using Morris-type recurrence formulas for the Lusztigq-analogues [13], we prove that this is also true for the coefficientsK_λ,μ^g,k whereμis a fixed nonempty partition. We also observe that these formulas permit to give an explicit lower bound for the degree of theq-analoguesK_λ,μ^g (q)such thatK_λ,μ^g (q)=0. We establish that the limits of the coefficientsK_λ,μ^so2n+1,kandK_λ,μ^so2n,kare the same. WriteK_λ,μ^so,kandK_λ,μ^sp,krespectively for the limits of the coefficientsK_λ,μ^so2n+1,kandK_λ,μ^sp2n,kwhenntends to infinity.

The stabilization property of the coefficientsK_λ,μ^g,k suggests then to introduce the formal series

K_λ,μ^so(q)=

k≥0

K_λ,μ^so,kq^k and K_λ,μ^sp(q)=

k≥0

K_λ,μ^sp,kq^k.

These series belong toN[[q]]and can be regarded as natural limits of the polynomials K_λ,μ^g (q). As far as the author is aware, the first occurrence of such limits for the Lusztigq-analogues was in [3] in the particular caseμ= ∅. We establish a duality between the formal seriesK_λ,∅^so(q)andK_λ,∅^sp(q)(Theorem5.3.1). Namely, we have

K_λ,^so_∅(q)=K_λ^sp_,∅(q) (3) whereλ is the conjugate partition ofλ. Note that (3) do not hold in general if we replace the formal seriesK_λ,μ^so(q)andK_λ,μ^sp(q)by the polynomialsK_λ,μ^g,k(q). We also give recurrence formulas (39), (40) for the seriesK_λ,μ^so(q)andK_λ,μ^sp(q)which permit efficient recursive computations. Thanks to these recurrence formulas, one derives simple expressions for the formal series K_λ,μ^so(q) andK_λ,μ^sp(q) whenλ is a single row or a single column partition (Proposition5.4.1). Such simple formulas seem not to exist for the Lusztig q-analogues K_λ,μ^g (q)even in the cases when λ is a single column or a single row partition. Moreover the duality (3) is false in general for the

polynomialsK_λ,μ^g (q). This suggests that the study of the seriesK_λ,μ^so(q)andK_λ,μ^sp(q) which is initiated in this paper, could be easier than that of the Lusztigq-analogues.

In this paper we restrict ourselves to the polynomial representations ofSO_N(C) andSp_N(C). Nevertheless, our results can also be extended to the q-analogues of weight multiplicities corresponding to the irreducible finite-dimensional rational rep- resentations ofGL_n(C). Limits of suchq-analogues in large rank for the zero weight have been previously studied, notably by Gupta [2] and Hanlon [3]. They can be ob- tained from the Cauchy formula which then plays the role of the Littlewood identities used in Section3. The existence of such limits for any dominant weight is then de- duced from the Morris recurrence formula for the Lusztigq-analogues associated to the root systems of typeAby using arguments similar to those of Sections4and5.

This also yields the stabilization of the dimensions in the Brylinski-Kostant filtration and recurrence formulas for the limits obtained.

The paper is organized as follows. In Section 2 we recall the necessary back- ground on symplectic and orthogonal Lie algebras, universal characters, and Lusztig q-analogues which is needed in the sequel. In Section3we introduce universal graded characters as limits in infinitely many variables for the graded characters associated toS(g)andH (g). We obtain the stabilization property of the coefficients K_λ,μ^g,k in Section4and reformulate this result in terms of the Brylinski-Kostant filtration. In Section5, we introduce the formal seriesK_λ,μ^so(q)andK_λ,μ^sp(q), establish recurrence formulas which permit to compute them by induction, prove the duality (3) and give explicit formulas forK_λ,μ^so(q)andK_λ,μ^sp(q)whenλis a row or a column partition.

2 Background

2.1 Convention for the root systems of typesB, CandD

In the sequelGis one of the complex Lie groupsSp2n, SO2n+1orSO2nandgis its Lie algebra. We follow the convention of [9] to realizeGas a subgroup ofGLN and gas a subalgebra ofgl_Nwhere

N=

⎧⎨

⎩

2n whenG=Sp_2n, 2n+1 whenG=SO_2n+1, 2n whenG=SO_2n.

With this convention the maximal torus T of G and the Cartan subalgebra hof g coincide respectively with the subgroup and the subalgebra of diagonal matrices of G andg. Similarly the Borel subgroup B of G and the Borel subalgebrab₊ of g coincide respectively with the subgroup and subalgebra of upper triangular matrices of Gandg. This gives the triangular decompositiong=b₊⊕h⊕b₋ for the Lie algebra g. Let e_i, h_i, f_i, i∈ {1, . . . , n}be a set of Chevalley generators such that e_i∈b₊, h_i∈handf_i∈b₋for anyi.

Letd_Nbe the linear subspace ofgl_N consisting of the diagonal matrices. For any i∈ {1, . . . , n}, writeε_i for the linear mapε_i:d_N→Csuch thatε_i(D)=δ_i for any diagonal matrixDwhose(i, i)-coefficient isδ_i. Then(ε₁, . . . , ε_n)is an orthonormal

basis of the Euclidean spaceh^∗_R(the real part ofh^∗). We denote by<·,·>the usual scalar product onh^∗_R. For anyβ∈h^∗_R, we writeβ=(β₁, . . . , β_n)for the coordinates ofβon the basis(ε₁, . . . , ε_n).

LetRbe the root system associated toG. We can take for the simple roots ofg

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

+= {α_n=ε_nandα_i=ε_i−ε_i+1, i=1, . . . , n−1}

for the root systemB_n,

+= {α_n=2εnandα_i=ε_i−ε_i+1, i=1, . . . , n−1}

for the root systemC_n,

+= {α_n=ε_n+ε_n−1andα_i=ε_i−ε_i+1, i=1, . . . , n−1}

for the root systemD_n.

(4)

Then the sets of positive roots are

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

R⁺= {εi−εj, εi+εj with 1≤i < j≤n} ∪ {εi with 1≤i≤n} for the root systemBn,

R⁺= {ε_i−ε_j, ε_i+ε_j with 1≤i < j≤n} ∪ {2εi with 1≤i≤n} for the root systemC_n,

R⁺= {ε_i−ε_j, ε_i+ε_j with 1≤i < j≤n} for the root systemD_n.

We denote byR the set of roots ofG. For any α∈R, letα^∨= <α,α>^α be the co- root corresponding toα. The Weyl group of the Lie groupGis the subgroup of the permutation group of the set{n, . . . ,2,1,1,2, . . . , n}generated by the permutations

⎧⎪

⎪⎨

⎪⎪

⎩

si=(i, i+1)(i, i+1), i=1, . . . , n−1 andsn=(n, n) for the root systemsBnandCn,

s_i=(i, i+1)(i, i+1), i=1, . . . , n−1 ands_n =(n, n−1)(n−1, n) for the root systemD_n

where fora=b (a, b)is the simple transposition which switchesaandb. We identify the subgroup ofW^ggenerated bys_i=(i, i+1)(i, i+1),i=1, . . . , n−1 with the symmetric groupS_n. We denote bythe length function corresponding to the above set of generators. The action ofw∈W^gonβ=(β₁, . . . , β_n)∈h^∗_Ris defined by

w·(β₁, . . . , β_n)=(β₁^w−1, . . . , β_n^w−1)

whereβ_i^w=β_w(i)ifw(i)∈ {1, . . . , n}andβ_i^w= −β_w(i)otherwise. We denote byρ the half sum of the positive roots ofR⁺. The dot action ofW^gonβ=(β₁, . . . , β_n)∈ h^∗_Ris defined by

w◦β=w·(β+ρ)−ρ. (5)

WriteP andP⁺for the weight lattice and the cone of dominant weights ofG. As usual we consider the order onP defined byβ≤γ if and only ifγ−β∈Q⁺.

For any positive integer m, denote byPm the set of partitions with at most m nonzero parts. LetPm(k), k∈Nbe the subset ofPmconsisting of the partitionsλ such that|λ| =λ₁+ · · · +λ_m=k. SetP= ∪m∈NPmandPm[k] = ∪a≤kPm(a).

Each partitionλ=(λ₁, . . . , λ_n)∈Pn can be identified with the dominant weight _n

i=1λ_iε_i. Then the irreducible finite-dimensional polynomial representations of SO2n+1andSp2n are parametrized by the partitions of Pn. The irreducible finite- dimensional polynomial representations ofSO_2n are parametrized by the dominant weights ofPn∪Pnwhere

Pn= {(λ1, . . . , λn−1,−λn)∈Zⁿ|(λ1, . . . , λn−1, λn)∈Pn}. For anyλ∈Pn, we denote byV^g(λ)

– the irreducible finite-dimensional representation of Gcorresponding to λ when g=so_2n+1,sp_2nor wheng=so_2nandλ_n=0,

– the direct sum of the representations of SO2n corresponding to the dominant weightsλandλ=(λ₁, . . . , λ_n−1,−λ_n)wheng=so_2nandλ_n=0.

The representationV^g(1)associated to the partitionλ=(1)is called the vector representation ofG. For any weightβ∈P and any partitionλ∈Pn, we writeV^g(λ)_β for the weight space associated toβ inV^g(λ).

We denote byQthe root lattice ofgand writeQ⁺for the elements ofQwhich are linear combination of positive roots with nonnegative coefficients.

The exponents {m₁, . . . , m_n} of the root system R verifies m_i =2i−1, i = 1, . . . , nwhenRis of typeB_norC_nand

m_i=2i−1, i∈ {1, . . . , n−1} and m_n=n−1 (6) whenRis of typeD_n.

Remark

(i) The integer n−1 appears twice in the exponents of a root system of typeD_n whennis even.

(ii) The exponents mi, i=1, . . . , n−1 are the same for the three root systems of typeB_n, C_norD_n.

As customary, we identify the latticeP of weights ofGwith a sublattice of(¹₂Z)ⁿ. For any β=(β1, . . . , β_n)∈P, we set|β| =β1+ · · · +β_n. We use for a basis of the group algebra Z[Zⁿ], the formal exponentials(e^β)_β∈Zⁿ satisfying the relations e^β1e^β2=e^β1+β2. We furthermore introducenindependent indeterminatesx1, . . . , xn

in order to identifyZ[Zⁿ]with the ring of polynomialsZ[x₁, . . . , x_n, x₁⁻¹, . . . , x_n⁻¹] by writinge^β=x₁^β1· · ·x_n^βn=x^βfor anyβ=(β1, . . . , β_n)∈Zⁿ.

Write s_λ^gln for the Weyl character (Schur function) of the finite-dimensional gl_n-module V^gln(λ) of highest weight λ. The character ring of GL_n is _n = Z[x1, . . . , xn]^symthe ring of symmetric functions innvariables.

For any λ∈Pn, we denote by s_λ^g the Weyl character ofV^g(λ). LetR^g be the character ring ofG. Then

R^g=Z[x1, . . . , x_n, x₁⁻¹, . . . , x⁻_n¹]^Wso2n+1 is theZ-algebra with basis{s_λ^g|λ∈Pn}.

In the sequel we will supposen≥2 when g=sp_2n or so_2n+1andn≥4 when g=so2n.

For each Lie algebra g=so_N or sp_N and any partitionν∈PN, we denote by V^glN(ν)↓^glg^N the restriction ofV^glN(ν)tog. Set

V^glN(ν)↓^glso^N_N =

λ∈Pn

V^soN(λ)^⊕b

soN ν,λ ,

(7) V^gl2n(ν)↓^glsp²ⁿ_2n=

λ∈Pn

V^sp2n(λ)^⊕b

sp2n ν,λ .

These formulas define in particular the branching coefficientsb^so_ν,λ^N andb_ν,λ^sp2n. The restriction mapr^gis defined by setting

r^g:

Z[x₁, . . . , x_N]^sym→R^g,

s_ν^glN−→char(V^glN(ν)↓^glg^N)

where char(V^glN(ν)↓^glg^N)is the character of theg-moduleV^glN(ν)↓^glg^N. We have then

r^g(s_ν^glN)=

s_ν^glN(x₁, . . . , x_n, x_n⁻¹, . . . , x₁⁻¹) whenN=2n, s_ν^glN(x1, . . . , x_n,1, x_n⁻¹, . . . , x₁⁻¹) whenN=2n+1.

LetPn⁽²⁾andPn^(1,1)be the subsets ofPnconsisting of the partitions with even length rows and the partitions with even length columns, respectively. Whenν∈Pnwe have the following formulas for the branching coefficientsb^so_ν,λ^N andb^sp_ν,λ²ⁿ:

Proposition 2.1.1 (see [10] appendix p. 295) Considerν∈Pn. Then:

1. b^so_ν,λ²ⁿ⁺¹=b_ν,λ^so2n=

γ∈Pn⁽²⁾c^ν_λ,γ 2. b^sp_ν,λ²ⁿ=

γ∈Pn^(1,1)c_λ,γ^ν

wherec^ν_{γ ,λ}is the usual Littlewood-Richardson coefficient corresponding to the parti- tionsγ , λandν.

Note that the equalityb^so_ν,λ²ⁿ⁺¹=b_ν,λ^so2nbecomes false in general whenν /∈Pn. As suggested by Proposition2.1.1, the manipulation of the Weyl characters is sim- plified by working with infinitely many variables. In [8], Koike and Terada have intro- duced a universal character ring for the classical Lie groups. This ring can be regarded as the ring=Z[x₁, . . . , x_n, . . .]^symof symmetric functions in countably many vari- ables. It is equipped with three naturalZ-bases indexed by partitions, namely

B^gl= {s^gl_λ |λ∈P}, B^sp= {s^sp_λ |λ∈P}, B^so= {s^so_λ |λ∈P}. (8)

We have in particular the decompositions:

s^gl_ν =

λ∈P

γ∈Pn⁽²⁾

c^ν_λ,γs^so_λ and s^gl_ν =

λ∈P

γ∈Pn^(1,1)

c^ν_λ,γs^sp_λ . (9)

We denote byϕthe linear involution defined onbyϕ(s^gl_λ )=s^gl_λ. Then, one has

ϕ(s^so_λ )=s^sp_λ. (10)

For any positive integern, denote by_n=Z[x₁, . . . , x_n]^symthe ring of symmetric functions innvariables. Write

π_n:Z[x₁, . . . , x_n, . . .]^sym→Z[x₁, . . . , x_n]^sym

for the ring homomorphism obtained by specializing each variable x_i, i > nat 0.

Thenπ_n(s^gl_λ )=s_λ^gln. Letπ^sp2nandπ^soN be the specialization homomorphisms de- fined by settingπ^sp2n=r^sp2n◦π_2nandπ^soN =r^soN◦π_N. For any partitionλ∈Pn

one has

s_λ^sp2n=π^sp2n(s^sp_λ ) and s_λ^soN=π^soN(s^s0_λ ).

We shall also need the following proposition (see [8] Corollary 2.5.3).

Proposition 2.1.2 Consider a Lie algebragof typeX_n∈ {B_n, C_n, D_n}. Letλ∈Pr

andμ∈Ps. Supposen≥r+sand set V^g(λ)⊗V^g(μ)=

ν∈Pn

V^g(ν)^⊕dλ,μν .

Then the coefficientsd_λ,μ^ν neither depend on the rank n ofgnor on its typeB, C orD.

Remark The previous proposition follows from the decompositions s^sp_λ ×s^sp_μ =

ν∈P

d_λ,μ^ν s^sp_ν and s^so_λ ×s^so_μ =

ν∈P

d_λ,μ^ν s^so_ν

for anyλ, μ∈P, in the ring. 2.2 Lusztigq-analogues

Theq-analoguePq^gof the Kostant partition function associated to the root systemR of the Lie algebragis defined by the equality

α∈R⁺

1

1−qe^α =

β∈Zⁿ

Pq^g(β)e^β.

Note thatPq^g(β)=0 if β /∈Q⁺. Givenλandμ two partitions ofPn, the Lusztig q-analogues of weight multiplicity is the polynomial

K_λ,μ^g (q)=

w∈W^g

(−1)^(w)Pq^g(w◦λ−μ).

It follows from the Weyl character formula thatK_λ,μ^g (1)is equal to the dimension of V^g(λ)μ.

Theorem 2.2.1 (Lusztig [14])

For any partitions λ, μ∈Pn, the polynomial K_λ,μ^g (q) has nonnegative integer coefficients.

We write

K_λ,μ^g (q)=

k≥0

K_λ,μ^g,kq^k. (11)

Then

K_λ,μ^g,k(q)=

w∈W^g

(−1)^(w)P^k(w◦λ−μ) (12) where for anyβ∈Zⁿ,P^k(β)is the number of ways of decomposingβ as a sum ofk positive roots.

Remark One easily verifies that K_λ,μ^g (q)=0 only if λ ≥ μ. Moreover, when

|μ| = |λ|, one hasK_λ,μ^g (q)=K_λ,μ^gln(q)whereK_λ,μ^gln(q)is the Kostka polynomial as- sociated to(λ, μ), i.e. the Lusztigq-analogue associated to the partitionsλ, μfor the root systemAn−1.

We also introduce the Hall-Littlewood polynomialsQμ^g,μ∈Pndefined by Q_μ^g=

λ∈Pn

K_λ,μ^g (q)s_λ^g.

2.3 The symmetric algebraS(g)

Considered as aG-module,gis irreducible and we have

⎧⎨

⎩

so2n+1V^so2n+1(1,1) and dim(so2n+1)=n(2n+1), sp_2nV^sp2n(2) and dim(sp_2n)=n(2n+1),

so_2nV^so2n(1,1) and dim(so2n)=n(2n−1).

(13)

LetS(g)be the symmetric algebra ofgand set S(g)=

k≥0

S^k(g)

whereS^k(g)is thek-th symmetric power ofg. By Proposition2.1.1and (13), we have

so_NV^glN(1,1)↓^glso^N_N and sp_2nV^gl2n(2)↓^glsp²ⁿ_2n. This implies the following isomorphisms

S^k(soN)S^k(V^soN(1,1))↓^glso^N_N and S^k(sp_2n)S^k(V^sp2n(2))↓^glsp²ⁿ_2n (14) for any nonnegative integerk.

Example 2.3.1 By using the Weyl dimension formula (see [4] page 303), one easily obtains the decompositions

S²(V^glN(1,1))V^glN(1,1,1,1)⊕V^glN(2,2) and

S²(V^gl2n(2))V^gl2n(4)⊕V^gl2n(2,2).

Hence by (14) and Proposition2.1.1, this gives

S²(g)V^g(1,1,1,1)⊕V^g(2,2)⊕V^g(2,0)⊕V^g(∅)forg=so_N and

S²(sp_2n)V^sp2n(4)⊕V^sp2n(2,2)⊕V^sp2n(1,1)⊕V^sp2n(∅).

Remark By the previous formulas, the multiplicities appearing in the decomposition of the square symmetric power of the Lie algebragof typeX_n∈ {B_n, C_n, D_n}do not depend on its rank providingn≥2. We give in Proposition3.1.1, the general explicit decomposition ofS^k(g)into its irreducible components.

3 Graded characters

3.1 Graded character of the symmetric algebra

LetV be aG orGL_n-module. For any nonnegative integer k, writeS^k(V )for the k-th symmetric power ofV and setS(V )= ⊕k≥0S^k(V ). ThenS^k(V )andS(V )are alsoG-modules. The graded character ofS(V )is defined by

charq(S(V ))=

k≥0

char(S^k(V ))q^k.

Denote byW(V )the collection of weights of the moduleV counted with their mul- tiplicities. Then we have

charq(S(V ))=

β∈W(V )

1 1−qe^β.

The weights of the Lie algebragof ranknconsidered as aG-module are such that W(g)= {α∈R,0, . . . , 0

ntimes

}.

Thus the graded character charq(S(g))ofS(g)verifies charq(S(g))= 1

(1−q)ⁿ

α∈R

1

1−qx^α. (15)

Proposition 3.1.1 For any nonnegative integerk, we have

char(S^k(so_N))=

λ∈Pn

ν∈PN^(1,1)(2k)

b_ν,λ^soNs_λ^soN,

char(S^k(sp_2n))=

λ∈Pn

ν∈P2n⁽²⁾(2k)

b^sp_ν,λ²ⁿs_λ^sp2n,

whereb^so_ν,λ^N andb_ν,λ^sp2nare the branching coefficients defined in (7).

Proof Suppose firstg=sp_2n. Recall the classical identity

1≤i≤j≤2n

1

1−x_ix_j =

ν∈P2n⁽²⁾

s_ν^gl2n

due to Littlewood. It immediately implies the decomposition

1≤i≤j≤2n

1

1−qxixj =

ν∈P2n⁽²⁾

q^|ν|2s_ν^gl2n=

k≥0

ν∈P2n⁽²⁾(2k)

s^gl_ν ²ⁿq^k.

By applying the restriction mapr^sp2n, this gives 1

(1−q)ⁿ

1≤i<j≤n

1 1−q_x^xi

j

1 1−q^x_x^j

i

1≤r≤s≤n

1 1−qxrxs

1 1−q_x¹

rxs

=

k≥0

ν∈P2n⁽²⁾(2k)

s_ν^gl2n(x1, . . . , x_n, x_n⁻¹, . . . , x⁻₁¹)q^k.

From (7), this can be rewritten in the form charq(S(sp_2n))= 1

(1−q)ⁿ

α∈R

1

1−qx^α =

k≥0

λ∈Pn

ν∈P2n⁽²⁾(2k)

b^sp_ν,λ²ⁿs_λ^sp2nq^k

which gives the desired identity by considering the coefficient inq^k.

Wheng=so_2n+1org=so_2n, one uses the identity

1≤i<j≤2n

1

1−qxixj =

ν∈P2n^(1,1)

q^|ν|2s_ν^gl2n=

k≥0

ν∈P2n^(1,1)(2k)

s_ν^gl2nq^k

and our result follows by similar arguments.

In the sequel, we set m^so_k,λ^N=

ν∈PN^(1,1)(2k)

b^so_ν,λ^N and m^sp_k,λ²ⁿ=

ν∈P2n⁽²⁾(2k)

b^sp_ν,λ²ⁿ.

Thus we have char(S^k(soN)) =

λ∈Pnm^so_k,λ^Ns_λ^soN and char(S^k(sp_2n)) =

λ∈Pnm^sp_k,λ²ⁿs_λ^sp2n.

3.2 Universal graded characters charq(S(sp))and charq(S(so))

Proposition 3.2.1 Consider a nonnegative integer k and supposen≥2k. For any λ∈P, we have the identities

m^so_k,λ²ⁿ⁺¹=m^so_k,λ²ⁿ=

ν∈P^(1,1)(2k)

γ∈P⁽²⁾

c^ν_λ,γ,

(16) m^sp_k,λ²ⁿ=

ν∈P⁽²⁾(2k)

γ∈P^(1,1)

c^ν_λ,γ.

In particular, the multiplicitiesm^sp_k,λ²ⁿ, m^so_k,λ²ⁿ⁺¹, m^so_k,λ²ⁿdo not depend onn.

Proof For anyν∈PN(2k) we haveν∈PN[n]sincen≥2k. In particularν∈Pn. We can thus deduce from Propositions2.1.1and3.1.1the decompositions

m^so_k,λ^N=

ν∈Pn^(1,1)(2k)

γ∈P2k⁽²⁾

c^ν_λ,γ andm^sp_k,λ²ⁿ=

ν∈Pn⁽²⁾(2k)

γ∈P2k^(1,1)

c^ν_λ,γ.

Sincec_λ,γ^ν =0 when|λ| + |γ| =2k(so|λ| ≤2k≤nand|γ| ≤n),m^so_k,λ^N andm^sp_k,λ²ⁿ can be rewritten as in (16) and thus, do not depend onn.

We set m^so_k,λ= lim

n→∞m^so_k,λ²ⁿ⁺¹ =

ν∈P^(1,1)(2k)

γ∈P^(1,1)

c_λ,γ^ν and m^sp_k,λ= lim

n→∞m^sp_k,λ²ⁿ

=

ν∈P⁽²⁾(2k)

γ∈P⁽²⁾

c_λ,γ^ν . (17)

Lemma 3.2.2 For any nonnegative integerkand any partitionλ,we have 1. m^sp_k,λ=m^so_k,λ

2. m^sp_k,λ=m^so_k,λ=0 if|λ|>2k

wherem^sp_k,λandm^so_k,λ are the multiplicities defined in (17).

Proof Write ι for the bijective map defined on P by λ−→λ. We then have ι(P⁽²⁾)=P^(1,1) andι(P⁽²⁾(2k))=P^(1,1)(2k). Moreoverc^ν_λ,γ =c_λ^ν_,γ for any par- titionsλ, γ , ν. This implies assertion 1 from the definition (17) ofm^sp_k,λandm^so_k,λ.

Recall thatc^ν_λ,γ =0 when|ν| = |λ| + |γ|. Since|ν| =2kin the equalities of (17),

we havem^sp_k,λ=m^so_k,λ=0 if|λ|>2k.

We define the universal graded characters charq(S(sp))and charq(S(so))by set- ting

charq(S(sp))=

1≤i≤j

1

1−qx_ix_j and charq(S(sp))=

1≤i<j

1

1−qx_ix_j. (18) Note that charq(S(sp))and charq(S(so))belong to the ring[[q]]of formal series with coefficients in.

For any F =

k≥0c_kq^k in [[q]], the specialization homomorphisms π^sp2n, π^so2n+1 andπ^so2nare then defined by setting

π^g(F )=

k≥0

π^g(c_k)q^k. (19)

By (15) and (18), we then have

π^soN(charq(S(so)))=charq(S(so_N)),

(20) π^sp2n(charq(S(sp)))=charq(S(sp_2n)).

Similarly, the linear involutionϕ(see (10)) is defined on[[q]]by ϕ(F )=

k≥0

ϕ(c_k)q^k. (21)

Proposition 3.2.3 We have the decompositions charq(S(so))=

k≥0

λ∈P

m^so_k,λs^so_λ q^k,

(22) charq(S(sp))=

k≥0

λ∈P

m^sp_k,λs^sp_λ q^k.

Proof One can write

charq(S(so))=

1≤i<j

1

1−qxixj =

k≥0

ν∈P^(1,1)(2k)

s^gl_ν q^k=

k≥0

λ∈P

m^so_k,λs^so_λ q^k

charq(S(sp))=

1≤i≤j

1

1−qxixj =

k≥0

ν∈P⁽²⁾(2k)

s^gl_ν q^k=

k≥0

λ∈P

m^sp_k,λs^sp_λ q^k

where the rightmost equalities follow from (9).

By using 1 of Lemma3.2.2, one derives the following corollary:

Corollary 3.2.4 We haveϕ(charq(S(so)))=charq(S(sp)).

Remark By 2 of Lemma3.2.2, one has char(S^k(so))=

λ∈P[2k]

m^so_k,λs^so_λ and char(S^k(sp))=

λ∈P[2k]

m^sp_k,λs^sp_λ (23)

whereP[2k]is the set of partitionsλsuch that|λ| ≤2k.

3.3 Universal graded character for harmonic polynomials

Letgbe a Lie algebra of typeX_n∈ {B_n, C_n, D_n}. Since the symmetric algebraS(g) can be regarded as aG-module, one can consider

S(g)^G= {x∈S(g)|g·x=xfor anyg∈G},

the ring of theG-invariants inS(g). By a classical theorem of Kostant [8] we have

S(g)=H (g)⊗S(g)^G (24)

whereH (g)is the ring ofG-harmonic polynomials. The ringS(g)^Gis generated by algebraically independent homogeneous polynomials of degreesd_i=m_i+1 and the graded character ofS(g)^Gconsidered as aG-module verifies

charq(S(g)^G)= n i=1

1 1−q^di. By (24) the graded character ofH (g)can be written

charq(H (g))= charq(S(g)) charq(S(g)^G)=

n i=1

(1−q^di)charq(S(g))=

k≥0

char(H^k(g))q^k. (25)