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

Lecouvey, Cédric

Journal of Algebraic Combinatorics, Tome 27 (2008) no. 4, pp. 451-477.

Résumé

Summary: Let $G$ be a simple complex classical Lie group with Lie algebra $\frak g$ frakg of rank $n$. We show that the coefficient of degree $k$ in the Lusztig $q$ -analogue $K _{ l, m} ^{\frak g}( q)$ K_$\lambda ,\mu $^frakg$(q)$ associated to the fixed partitions $\lambda $ and $\mu $ 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 $( \lambda , \mu )$, 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 $\lambda $ is a row or a column partition.

Détail

Keywords: keywords Brylinski filtration, $q$-analogues, Lusztig polynomials, $q$-series