Stabilization of the Brylinski-Kostant filtration and limit of Lusztig q -analogues
Cédric Lecouvey
Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville B.P. 699 62228 Calais Cedex France
DOI: 10.1007/s10801-007-0097-9
Abstract
Let G be a simple complex classical Lie group with Lie algebra \frak g \frak{g} of rank n. We show that the coefficient of degree k in the Lusztig q -analogue K l, m \frak g( q) K_{λ,μ}^{\frak{g}}(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.
Pages: 451–477
Keywords: keywords Brylinski filtration; $q$-analogues; Lusztig polynomials; $q$-series
Full Text: PDF
References
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2. Gupta, R.K.: Generalized exponents via Hall-Littlewood symmetric functions. Bull. Am. Math. Soc. 16(2), 287-291 (1987)
3. Hanlon, P.: On the decomposition of the tensor algebra of the classical Lie algebras. Adv. Math. 56, 238-282 (1985)
4. Goodman, G., Wallach, N.R.: Representation Theory and Invariants of the Classical Groups. Cambridge University Press, Cambridge (2003)
5. Hesselink, W.-H.: Characters of the nullcone. Math. Ann. 252, 179-182 (1980)
6. Kashiwara, M., Nakashima, T.: Crystal graphs for representations of the q-analogue of classical Lie algebras. J. Algebra 165, 295-345 (1994)
7. Kato, S.: Spherical functions and a q-analogue of Kostant's weight multiplicity formula. Inv. Math. 66, 461-468 (1982)
8. Koike, K., Terada, I.: Young diagrammatic methods for the representation theory of the classical groups of type Bn, Cn and Dn. J. Algebra 107, 466-511 (1987)
9. Koike, K., Terada, I.: Young diagrammatic methods for the restriction of representations of complex classical Lie groups to reductive subgroups of maximal rank. Adv. Math. 79, 104-135 (1990)
10. Littlewood, D.-E.: The Theory of Group Characters and Matrix Representations of Groups, 2nd edn.