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

Let G be a simple complex classical Lie group with Lie algebra g of rank n. We show that the coefficient of degree k in the Lusztig q-analogue K λ,μ(q) associated to the fixed partitions λ 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 (λ,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 λ is a row or a column partition.