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Let G be a simple complex classical Lie group with Lie algebra $\frak{g}$ of rank n. We show that the coefficient of degree k in the Lusztig q -analogue $K_{\lambda ,\mu }^{\frak{g}}(q)$ associated to the fixed partitions 驴 and μ stabilizes for n sufficiently large. As a consequence, we obtain the stabilization of the dimensions in the Brylinski-Kostant filtration associated to any dominant weight. We then introduce, for each pair of partitions (驴,μ), formal series which can be regarded as natural limits of the Lusztig q-analogues. We give a duality property for these limits and recurrence formulas which permit notably to derive explicit expressions when 驴 is a row or a column partition.

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