Universal graded characters and limit of Lusztig $q$-analogues

Abstract

Let G be a symplectic or orthogonal complex Lie group with Lie algebra g. As a G-module, the decomposition of the symmetric algebra S(g) into its irreducible components can be explicitely obtained by using identities due to Littlewood. We 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 are 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 by Koike and Terada. From Hesselink expression of the graded character of harmonics, the coefficient of degree k in the Lusztig q-analogue associated to the partition lambda and the weight 0 thus stablizes for n sufficiently large. By using Morris-type recurrence formulas, we prove that this is also true for the polynomials $K_{\lambda ,\mu}^{g}(q)$ where mu is a nonempty fixed partition. This can be reformulated in terms of a stability property for the dimension of the components of the Brylinski-Kostant filtration. We also associate to each pair of partitions (lambda, mu) formal series $K_{\lambda ,\mu}^{so}(q)$ and $K_{\lambda ,\mu}^{sp}(q)$, which can be regarded as natural limit of the Lusztig q-analogues. One gives a duality property for these limits and obtains simple expressions when lambda is a row or a column partition.