Table Of Content

Universal graded characters and limit of Lusztig q-analogues 7 Cédric Lecouvey 0 0 Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville 2 B.P. 699 62228 Calais Cedex n [email protected] a J 3 2 Abstract ] Let G be a symplectic or orthogonalcomplex Lie groupwith Lie algebrag. As a G-module, T the decomposition of the symmetric algebra S(g) into its irreducible components can be ex- R plicitelyobtainedbyusingidentitiesduetoLittlewood. Weshowthatthemultiplicitiesappear- . h inginthedecompositionofthek-thgradedcomponentofS(g)donotdependontheranknofg t providing n is sufficiently large. Thanks to a classical result by Kostant, we establish a similar a m result for the k-th graded component of the space H(g) of G-harmonic polynomials. These stabilizationproperties are equivalentto the existence of a limit in infinitely many variables for [ the graded characters associated to S(g) and H(g). The limits so obtained are formal series 1 with coefficients in the ring of universal characters introduced by Koike and Terada. v From Hesselink expression of the graded character of harmonics, the coefficient of degree 3 4 k in the Lusztig q-analogue Kλg,∅(q) associated to the fixed partition λ thus stablizes for n 6 sufficiently large. By using Morris-type recurrence formulas, we prove that this is also true 1 for the polynomials Kg (q) where µ is a nonempty fixed partition. This can be reformulated λ,µ 0 in terms of a stability property for the dimension of the components of the Brylinski-Kostant 7 filtration. We also associate to each pair of partitions (λ,µ) formal series Kso (q) and Ksp (q), 0 λ,µ λ,µ / which canbe regardedas naturallimit of the Lusztig q-analogues.One givesa duality property h for these limits and obtains simple expressions when λ is a row or a column partition. t a m 1 Introduction : v i X The multiplicity K of the weight µ in the irreducible finite-dimensional representation Vg(λ) of λ,µ r thesimpleLiegroupG withLiealgebra g can bewritten in terms of theordinaryKostant partition a function P defined by the equality 1 = P(β)eβ (1−eα) αpositiveroot β Y X where β runs on the set of nonnegative integral combinations of positive roots of g. Thus P(β) is the number of ways the weight β can be expressed as a sum of positive roots. Then, one derives from the Weyl character formula Kg = (−1)ℓ(w)P(w(λ+ρ)−(µ+ρ)) (1) λ,µ w∈Wg X where Wg is the Weyl group of g. 1 g The Lusztig q-analogue of weight multiplicity K (q) is obtained by substituting the ordinary λ,µ Kostant partition function P by its q-analogue P in (1). Namely P is defined by the equality q q 1 = P (β)eβ (1−qeα) q αpositiveroot β Y X and we have Kg (q) = (−1)ℓ(w)P (w(λ+ρ)−(µ+ρ)). λ,µ q w∈Wg X g As shown by Lusztig [14], K (q) is a polynomial in q with nonnegative integer coefficients. Many λ,µ interpretations of the Lusztig q-analogues exist. In particular, they can be obtained from the g Brylinski-Kostant filtration of weight spaces [1]. The polynomials K (q) appear in the graded λ,∅ character of the harmonic polynomials associated to g [4]. We also recover the Lusztig q-analogues as the coefficients of the expansion of the Hall-Littlewood polynomials on the basis of Weyl characters (see [17]). This notably permits to prove that there are affine Kazhdan-Lusztig polynomials. gl In [11] Lascoux and Schu¨tzenberger have obtained a combinatorial expression for K n(q) in terms λ,µ of thecharge statistic onthesemistandardtableaux of shapeλandevaluation µ.By usingthecom- binatorics of crystal graphs introduced by Kashiwara and Nakashima [5], 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. Consider λ,µ two partitions of length at most m.These partitions can be regarded as dominant gl weights for g = gl ,so ,sp or so when n≥ m. Then, K n(q) does not depend on the rankn n 2n+1 2n 2n λ,µ considered. SuchapropertydoesnotholdfortheLusztigq-analoguesKg (q)wheng = so ,sp λ,µ 2n+1 2n or so which depend in general on the rank of the Lie algebra considered. Write 2n Kg (q) = Kg,kqk. λ,µ λ,µ k≥0 X We first establish in this paper that for g = so ,sp or so , the coefficient Kg,k stabilizes 2n+1 2n 2n λ,µ when n tends to the infinity. More precisely, Kg,k does not depend on the rank n of g providing λ,µ n ≥ 2k + a where a is the number of nonzero parts of µ (Theorem 4.3.1). By Brylinski’s inter- g,k pretation of the coefficients K [1], one then obtains that the dimension of the k-th component λ,µ of the Brylinski-Kostant filtration associated to the finite-dimensional irreducible representations of g = so ,sp or so stabilizes for n sufficiently large (Theorem 4.4.2). Observe that this 2n+1 2n 2n gl stabilization is immediate for g = gl since the polynomials K n(q) does not depend on n. For n λ,µ g = so ,sp or so the Brylinski-Kostant filtration depends in general on the rank considered 2n+1 2n 2n and it seems difficult to obtain the dimension of its components by direct computations. Our methods 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 the k-th graded component of S(g) do not depend on the rank n of g providing n is sufficiently large. Thanks to a classical result by Kostant, we establish a similar result for the k-th graded component of the space H(g) of G-harmonic polynomials. These stabilization properties is equivalent to the existence of a limit in infinitely many variables for the graded characters associated to S(g) and H(g). The limits so obtained are formal series with coefficients in the ring of universal characters introduced 2 by Koike and Terada. From Hesselink expression [4] of the graded character of H(g), one then g,k derives that K stabilizes for n sufficiently large. By using Morris-type recurrence formulas for λ,∅ g,k the Lusztig q-analogues [13], we prove that this is also true for the coefficients K where µ is λ,µ a nonempty fixed partition. We also observe that these formulas permit to give an explicit lower g, g, boundfor the degree of the q-analogues K (q) such that K (q) 6= 0. We establish that the limits λ,µ λ,µ of the coefficients Kso2n+1,k and Kso2n+1,k are the same. Write Ksp,k and Kso,k respectively for the λ,µ λ,µ λ,µ λ,µ limits of the coefficients Kso2n+1,k and Ksp2n,k when n tends to the infinity. λ,µ λ,µ g,k The stabilization property of the coefficients K suggests then to introduce the formal series λ,µ Kso (q) = Kso,kqk and Ksp (q) = Ksp,kqk. λ,µ λ,µ λ,µ λ,µ k≥0 k≥0 X X g These series belong N[[q]] and can be regarded as natural limits of the polynomials K (q). We λ,µ establish a duality result between the formal series Kso(q) and Ksp(q) (Theorem 5.3.1). Namely, λ,∅ λ,∅ we have Kso(q) = Ksp (q) (2) λ,∅ λ′,∅ where λ′ is the conjugate partition of λ. Note that (2) do not hold in general if we replace the formal series Kso (q) and Ksop(q) by the polynomials Kg,k(q). We also give recurrence formulas λ,µ λ,µ λ,µ (38), (39) for the series Kso (q) and Ksop(q). Thanks to these recurrence formulas, one derives λ,µ λ,µ simple expressions for the formal series KX (q) when λ is a row or a column partition (Proposition λ,∅ g 5.4.1). Note that we have not find so simple formulas for the Lusztig q-analogues K (q) even in λ,µ the cases when λ is a column or a row partition. Moreover the duality (2) is false in general for the polynomials Kg (q). This suggests that the study of the series Kso (q) and Ksp (q) which is λ,µ λ,µ λ,µ initiated in this paper, could be easier than that of the Lusztig q-analogues. Thepaperisorganized asfollows.InSection2werecallthenecessarybackgroundonsymplectic and orthogonal Lie algebras, universal characters, and Lusztig q-analogues which is needed in the sequel. In Section 3 we introduce universal graded characters as limit in infinitely many variables for the graded characters associated to S(g) and H(g). We obtain the stabilization property of g,k the coefficients K in Section 4 and reformulate this result in terms of the Brylinski-Kostant λ,µ filtration. In Section 5, we introduce the formal series Kso (q) and Ksop(q), establish recurrence λ,µ λ,µ formulas which permit to compute them by induction, prove the duality (2) and give explicit formulas for Kso (q) and Ksop(q) when λ is a row or a column partition. Finally in Section 6 we λ,µ λ,µ have added a few considerations on the possibility to define Hall-Littlewood polynomials from the formal series Kso (q) and Ksop(q). λ,µ λ,µ 2 Background 2.1 Convention for the root systems of types B,C and D In the sequel G is one of the complex Lie groups Sp ,SO or SO and g its Lie algebra. We 2n 2n+1 2n follow the convention of [7] to realize G as a subgroup of GL and g as a subalgebra of gl where N N 2n when G = Sp 2n N = 2n+1 when G = SO . 2n+1  2n when G = SO  2n  3 With this convention the maximal torus T of G and the Cartan subalgebra h of g coincide respectively with the subgroup and the subalgebra of diagonal matrices of G and g. Similarly the Borel subgroup B of G and the Borel subalgebra b of g coincide respectively with the subgroup + and subalgebra of upper triangular matrices of G and g. This gives the triangular decomposition g = b ⊕h⊕b for the Lie algebra g. Let e ,h ,f , i ∈ {1,...,n} be a set of Chevalley generators + − i i i such that e ∈ b ,h ∈ h and f ∈ b for any i. i + i i − Let d be the linear subspace of gl consisting of the diagonal matrices. For any i ∈ {1,...,n}, N N write ε for the linear map ε : d → C such that ε (D) = δ for any diagonal matrix D whose i i N i i (i,i)-coefficient is δ . Then (ε ,...,ε ) is an orthonormal basis of the Euclidean space h∗ (the real i 1 n R part of h∗). We denote by < ·,· > the usual scalar product on h∗. R Let R be the root system associated to G. We can take for the simple roots of g Σ+ = {α = ε and α =ε −ε , i = 1,...,n−1 for the root system B } n n i i i+1 n Σ+ = {α = 2ε and α = ε −ε , i= 1,...,n−1 for the root system C } . (3) n n i i i+1 n  Σ+ = {α = ε +ε and α = ε −ε , i = 1,...,n−1 for the root system D }  n n n−1 i i i+1 n Then the sets of positive roots are R+ = {ε −ε ,ε +ε with 1 ≤ i< j ≤ n}∪{ε with 1 ≤ i ≤ n} for the root system B i j i j i n R+ = {ε −ε ,ε +ε with 1 ≤ i< j ≤ n}∪{2ε with 1 ≤ i≤ n} for the root system C . i j i j i n  R+ = {ε −ε ,ε +ε with 1 ≤ i< j ≤ n} for the root system D  i j i j n We denote by R the set of roots of G. For any α ∈R, let α∨ = α be the coroot corresponding <α,α> to α. The Weyl group of the Lie group G is the subgroup of the permutation group of the set {n,...,2,1,1,2,...,n} generated by the permutations s = (i,i+1)(i,i+1), i = 1,...,n−1 and s = (n,n) for the root systems B and C i n n n s = (i,i+1)(i,i+1), i = 1,...,n−1 and s′ = (n,n−1)(n−1,n) for the root system D (cid:26) i n n where for a 6= b (a,b) is the simple transposition which switches a and b. We identify the subgroup of Wg generated by s = (i,i+1)(i,i+1), i= 1,...,n−1 with the symmetric group S . We denote i n by ℓ the length function corresponding to the above set of generators. The action of w ∈ Wg on β = (β ,...,β ) ∈ h∗ is defined by 1 n R w·(β ,...,β ) = (βw−1,...,βw−1) 1 n 1 n where βw = β if w(i) ∈ {1,...,n} and βw = −β otherwise. We denote by ρ the half sum of i w(i) i w(i) the positive roots of R+. The dot action of Wg on β = (β ,...,β ) ∈ h∗ is defined by 1 n R w◦β = w·(β +ρ)−ρ. (4) Write P and P+ for the weight lattice and the cone of dominant weights of G. As usualwe consider the order on P defined by β ≤ γ if and only if β−γ ∈ Q+. For any positive integer m, denote by P the set of partitions with at most m nonzero parts. Let m P (k),k ∈ N be the subset of P containing the partitions λ such that |λ| = λ +···+λ = k. m m 1 m Set P = ∪m∈NPm and Pm[k] = ∪a≤kPm(a). Eachpartitionλ = (λ ,...,λ ) ∈P willbeidentifiedwiththedominantweight n λ ε .Thenthe 1 n n i=1 i i irreducible finite-dimensional polynomial representations of G are parametrized by the partitions P 4 of P . For any λ ∈ P , denote by Vg(λ) the irreducible finite-dimensional representation of G of n n highest weight λ. The representation Vg(1) associated to the partition λ = (1) is called the vector representation of G. For any weight β ∈ P and any partition λ ∈ P , we write Vg(λ) for the n β weight space associated to β in Vg(λ). WedenotebyQtherootlatticeofgandwriteQ+ fortheelementsofQwhicharelinearcombination of positive roots with nonnegative coefficients. The exponents {m ,...,m } of the root system R verifies m = 2i−1, i = 1,...,n when R is of type 1 n i B or C and n n m = 2i−1,i ∈ {1,...,n−1} and m = n−1 (5) i n when R is of type D . n Remarks: (i) : The integer n−1 appears twice in the exponents of a root system of type D when n is even. n (ii) : The exponents m ,i = 1,...,n−1 are the same for the three root systems of type B ,C or i n n D . n As customary, we identify P the lattice of weights of G with a sublattice of (1Z)n. For any β = 2 (β ,...,β ) ∈ P, we set |β| = β +···+ β . We use for a basis of the group algebra Z[Zn], the 1 n 1 n formal exponentials (eβ)β∈Zn satisfying the relations eβ1eβ2 = eβ1+β2. We furthermore introduce n independent indeterminates x ,...,x in order to identify Z[Zn] with the ring of polynomials 1 n Z[x1,...,xn,x−11,...,x−n1] by writing eβ = xβ11 ···xβnn = xβ for any β = (β1,...,βn)∈ Zn. Write sgln for the Weyl character (Schur function) of Vgln(λ) the finite-dimensional gl -module of λ n highestweight λ.Thecharacter ringofGL isΛ = Z[x ,...,x ]sym theringof symmetricfunctions n n 1 n in n variables. For any λ ∈ P , we denote by sg the Weyl character of Vg(λ). Let Rg be the character ring of G. n λ Then Rg= Z[x ,...,x ,x−1,...,x−1]Wg 1 n 1 n g is the Z-algebra with basis {s | λ ∈ P }. λ n In the sequel we will suppose n ≥ 2 when g = sp or so and n ≥ 4 when g = so . 2n 2n+1 2n For each Lie algebra g = soN or spN and any partition ν ∈ PN, we denote by VglN(ν) ↓gglN the restriction of VglN(ν) to g. Set VglN(ν) ↓ gglN = VsoN(λ)⊕bsνo,λN, (6) λM∈Pn Vgl2n(ν) ↓ ggl2n = Vso2n(λ)⊕bνsp,λ2n. λM∈Pn This definein particular the branching coefficients bsoN and bsp2n. The restriction map rgis defined ν,λ ν,λ by setting Z[x ,...,x ]sym → Rg rg: 1 N . (cid:26) sgνlN 7−→ char(VglN(ν)↓gglN) We have then sglN(x ,...,x ,x−1,...,x−1) when N = 2n rg(sglN) = ν 1 n n 1 . ν ( sgνlN(x1,...,xn,0,x−n1,...,x−11) when N = 2n+1 5 (2) (1,1) Let P and P be the subsets of P containing the partitions with even length rows and the n n n partitions with even length columns, respectively. When ν ∈P we have the following formulas for n the branching coefficients bsoN and bp2n : ν,λ ν,λ Proposition 2.1.1 (see [9] appendix p 295) Consider ν ∈P . Then: n 1. bso2n+1 = bso2n = cν ν,λ ν,λ γ∈Pn(2) λ,γ 2. bsp2n = Pcν ν,λ γ∈Pn(1,1) λ,γ where cν is tPhe usual Littlewood-Richardson coefficient corresponding to the partitions γ,λ and ν. γ,λ Note that the equality bso2n+1 = bso2n becomes false in general when ν ∈/ P . ν,λ ν,λ n Assuggested byProposition 2.1.1,themanipulation of theWeyl characters is simplifiedbyworking with infinitely many variables. In [6], Koike and Terada have introduced a universal character ring for the classical Lie groups. This ring can be regarded as the ring Λ = Z[x ,...,x ,...]sym of 1 n symmetricfunctionsincountably manyvariables.ItisequippedwiththreenaturalZ-basesindexed by partitions, namely Bgl={sgl |λ ∈P}, Bsp={ssp| λ ∈ P} and Bso={sso|λ ∈ P}. (7) λ λ λ We have in particular the decompositions: sgl= cν sso and sgl= cν ssp. (8) ν λ,γ λ ν λ,γ λ λX∈Pγ∈XPn(2) λX∈Pγ∈XPn(1,1) We denote by ϕ the linear involution defined on Λ by ϕ(sso)= ssp. (9) λ λ′ For any positive integer n, denote by Λ = Z[x ,...,x ]sym the ring of symmetric functions in n n 1 n variables. Write π : Z[x ,...,x ,...]sym → Z[x ,...,x ]sym n 1 n 1 n for the ring homomorphism obtained by specializing each variable x ,i > n at 0. Then π (sgl) = i n λ sgλln. Let πsp2n and πsoN be the specialization homomorphisms defined by setting πsp2n = rsp2n◦π2n and πsoN = rsoN ◦πN. For any partition λ ∈Pn one has ssp2n = πsp2n(ssp) and ssoN =πsoN(ss0). λ λ λ λ When λ ∈/ PN, we have πsp2n(ssλp) = 0 and πsoN(ssλ0) = 0. The situation is more complicated when λ ∈ P but λ ∈/ P , that is if d(λ) the number of parts of λ verifies n < d(λ) ≤N. In this case one N n shows by using determinantal identities for the Weyl characters (see [6] Proposition 2.4.1) that πsp2n(ssp) = ±ssp2n, πso2n+1(sso) = ±sso2n+1 and πso2n(sso)= ±sso2n λ ηC λ ηB λ ηD wherethesigns±andthepartitionsη ,η ,η aredeterminedbysimplecombinatorialprocedures. B C D Note that we have in this case |η |≤ |λ| for X = B,C,D. X We shall also need the following proposition (see [6] Corollary 2.5.3). 6 Proposition 2.1.2 Consider a Lie algebra g of type X ∈ {B ,C ,D }. Let λ ∈ P and µ ∈ P . n n n n r s Suppose n ≥ r+s and set Vg(λ)⊗Vg(µ)= Vg(ν)⊕dνλ,µ. νM∈Pn Then the coefficients dν do not depend on the rank n of g neither of its type B,C or D. λ,µ Remark: The previous proposition follows from the decompositions ssp×ssp= dν ssp and sso×sso= dν sso λ µ λ,µ ν λ µ λ,µ ν ν∈P ν∈P X X for any λ,µ ∈ P, in the ring Λ. 2.2 Lusztig q-analogues The q-analogue P of the Kostant partition function associated to the root system R of the Lie q algebra g is defined by the equality 1 = P (β)eβ. 1−qeα q αY∈R+ βX∈Zn Note that PBn(β) = 0 if β ∈/ Q+. Given λ and µ two partitions of P , the Lusztig q-analogues of q n weight multiplicity is the polynomial Kg (q) = (−1)ℓ(w)P (w◦λ−µ). λ,µ q w∈Wg X g g It follows from the Weyl character formula that K (1) is equal to the dimension of V (λ). λ,µ µ Theorem 2.2.1 (Lusztig [14]) g For any partitions λ,µ ∈ P , the polynomial K (q) has nonnegative integer coefficients. n λ,µ We write Kg (q) = Kg,kqk. (10) λ,µ λ,µ k≥0 X Then Kg,k(q) = (−1)ℓ(w)Pk(w◦λ−µ) (11) λ,µ w∈Wg X where for any β ∈ Zn, Pk(β) is the number of ways of decomposing β as a sum of k positive roots. Remark: One verifies easily that Kg (q) 6= 0 only if λ ≥ µ. Moreover, when |µ| = |λ|, one has λ,µ g gl gl K (q) = K n(q) where K n(q) is the Kostka polynomial associated to (λ,µ), i.e. the Lusztig λ,µ λ,µ λ,µ q-analogue associated to the partitions λ,µ for the root system A . n−1 ′g We also introduce the Hall-Littlewood polynomials Q , µ ∈ P defined by µ n Q′g= Kg (q)sg. µ λ,µ λ λX∈Pn 7 2.3 The symmetric algebra S(g) Considered as a G-module, g is irreducible and we have so ≃ Vg(1,1) and dim(so )= n(2n+1) 2n+1 2n+1 sp ≃ Vg(2) and dim(sp )= n(2n+1) . (12)  2n 2n so ≃ Vg(1,1) and dim(so ) = n(2n−1)  2n 2n Let S(g) be the symmetric algebra associated to g and set S(g) = Sk(g) k≥0 M where Sk(g) is the k-th symmetric power of g. By Proposition 2.1.1 and (12), we have g ≃ VglN(1,1) ↓gglN for g = soN and g ≃ Vgl2n(2) ↓gspl22nn for g = sp2n. This implies the following isomorphisms Sk(g)≃ Sk(VglN(1,1)) ↓gglN for g = soN and Sk(g)≃ Sk(Vgl2n(2)) ↓gspl22nn for g = sp2n. (13) for any nonnegative integer k. Example 2.3.1 By using the Weyl dimension formula (see [3] page 303), one easily obtain the decompositions S2(VglN(1,1)) ≃ VglN(1,1,1,1)⊕VglN(2,2) and S2(Vgl2n(2)) ≃ Vgl2n(4)⊕Vgl2n(2,2). Hence by 13 and Proposition 2.1.1, this gives S2(g) ≃ Vg(1,1,1,1)⊕Vg(2,2)⊕Vg(2,0)⊕Vg(∅) for g = so N and S2(sp ) ≃ Vsp2n(4)⊕Vsp2n(2,2)⊕Vsp2n(1,1)⊕Vsp2n(∅). 2n Remark: By the previous formulas, the multiplicities appearing in the decomposition of the square symmetric power of the Lie algebra g of type X ∈ {B ,C ,D } do not depend on its rank n n n n providing n ≥ 2. We give in Proposition 3.1.1, the general explicit decomposition of Sk(g) into its irreducible components. 3 Graded characters 3.1 Graded character of the symmetric algebra Let V be a G or GL -module. For any nonnegative integer k, write Sk(V) for the k-th symmetric n power of V and set S(V) = ⊕ Sk(V). Then Sk(V) and S(V) are also G-modules. The graded k≥0 character of S(V) is defined by char (S(V)) = char(Sk(V))qk. q k≥0 X 8 Denote by W(V) the collection of weight of the module V counted with their multiplicities. Then we have 1 char (S(V)) = . q 1−qeβ β∈W(V) Y The weights of the Lie algebra g of rank n considered as a G-module are such that W(g) = {α ∈ R,0,...,0}. ntimes Thus the graded character char (S(g)) of S(g) verifies| {z } q 1 1 char (S(g)) = . (14) q (1−q)n 1−qxα α∈R Y Proposition 3.1.1 For any nonnegative integer k, we have char (Sk(so )) = bsoNssoN q N ν,λ λ λX∈Pnν∈PX(1,1)(2k) N char (Sk(sp )) = bsp2nssp2n q 2n ν,λ λ λX∈Pnν∈PX(2)(2k) 2n where bsoN and bsp2n are the branching coefficients defined in (6). ν,λ ν,λ Proof. Suppose first g = sp . Recall the classical identity 2n 1 = sgl2n 1−x x ν i j 1≤iY≤j≤2n ν∈XP(2) 2n due to Littlewood. It immediately implies the decomposition 1−q1x x = q|ν2|sgνl2n = sgνl2nqk. i j 1≤iY≤j≤2n ν∈XP(2) Xk≥0ν∈PX(2)(2k) 2n 2n By applying the restriction map rsp2n, this gives 1 1 1 1 1 = (1−q)n 1−qxi 1−qxj 1−qx x 1−q 1 1≤iY<j≤n xj xi 1≤rY≤s≤n r s xrxs sνgl2n(x1,...,xn,x−n1,...,x−11)qk. Xk≥0ν∈PX(2)(2k) 2n From (6), this can be rewritten on the form 1 1 char (S(sp )) = = bsp2nssp2nqk q 2n (1−q)n 1−qxα ν,λ λ αY∈R Xk≥0λX∈Pnν∈PX(2)(2k) 2n 9 which gives the desired identity by considering the coefficient in qk. When g = so or g = so , one uses the identity 2n+1 2n 1−q1x x = q|ν2|sνgl2n = sgνl2nqk i j 1≤iY<j≤2n ν∈XP(1,1) Xk≥0ν∈PX(1,1)(2k) 2n 2n and our result follows by similar arguments. In the sequel, we set msoN = bsoN and msp2n = bsp2n. k,λ ν,λ k,λ ν,λ ν∈PX(1,1)(2k) ν∈PX(2)(2k) N 2n Thus we have char (Sk(so )) = msoNssoN and char (Sk(sp )) = msp2nssp2n. q N λ∈Pn k,λ λ q 2n λ∈Pn k,λ λ P P 3.2 Universal graded characters char (Ssp) and char (Sso) q q Proposition 3.2.1 Consider a nonnegative integer k and a partition λ ∈ P . Suppose n ≥ 2k. m Then we have the identities mso2n+1 = mso2n = cν (15) k,λ k,λ λ,γ ν∈PX(1,1)(2k)γ∈XP(2) msp2n = cν . k,λ λ,γ ν∈PX(2)(2k)γ∈XP(1,1) In particular, the multiplicities msp2n,mso2n+1 and mso2n do not depend on n. k,λ k,λ k,λ Proof. For any ν ∈ P (2k) we have ν ∈ P (n) since n ≥ 2k. We can thus deduce from N N Propositions 2.1.1 and 3.1.1 the decompositions msoN = cν and msp2n = cν . k,λ λ,γ k,λ λ,γ ν∈PnX(1,1)(2k)γ∈XP2(2k) ν∈PXn(2)(2k)γ∈XP2(1k,1) Since cν = 0 when |λ|+|γ| =6 2k, msoN and msp2n can be rewritten as in (15) and thus, do not λ,γ k,λ k,λ depend on n. We set mso = lim mso2n+1 = cν and msp = lim msp2n = cν . (16) k,λ n→∞ k,λ λ,γ k,λ n→∞ k,λ λ,γ ν∈PX(1,1)(2k)γ∈XP(1,1) ν∈PX(2)(2k)γ∈XP(2) Lemma 3.2.2 For any nonnegative integer k and any partition λ, we have 1. msp = mso k,λ k,λ′ 2. msp = mso = 0 if |λ|> 2k k,λ k,λ where msp and mso are the multiplicities defined in (16). k,λ k,λ′ 10