Abstract
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 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.
Article PDF
Similar content being viewed by others
References
Brylinski, R.-K.: Limits of weight spaces, Lusztig’s q-analogs and fiberings of adjoint orbits. J. Am. Math. Soc. 29(3), 517–533 (1989)
Gupta, R.K.: Generalized exponents via Hall-Littlewood symmetric functions. Bull. Am. Math. Soc. 16(2), 287–291 (1987)
Hanlon, P.: On the decomposition of the tensor algebra of the classical Lie algebras. Adv. Math. 56, 238–282 (1985)
Goodman, G., Wallach, N.R.: Representation Theory and Invariants of the Classical Groups. Cambridge University Press, Cambridge (2003)
Hesselink, W.-H.: Characters of the nullcone. Math. Ann. 252, 179–182 (1980)
Kashiwara, M., Nakashima, T.: Crystal graphs for representations of the q-analogue of classical Lie algebras. J. Algebra 165, 295–345 (1994)
Kato, S.: Spherical functions and a q-analogue of Kostant’s weight multiplicity formula. Inv. Math. 66, 461–468 (1982)
Koike, K., Terada, I.: Young diagrammatic methods for the representation theory of the classical groups of type B n ,C n and D n . J. Algebra 107, 466–511 (1987)
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)
Littlewood, D.-E.: The Theory of Group Characters and Matrix Representations of Groups, 2nd edn. Oxford University Press, Oxford (1958)
Lascoux, A., Schützenberger, M.-P.: Sur une conjecture de H.O. Foulkes. CR Acad. Sci. Paris 288, 95–98 (1979)
Lecouvey, C.: Kostka–Foulkes polynomials cyclage graphs and charge statistic for the root system C n . J. Algebr. Comb. 21, 203–240 (2005)
Lecouvey, C.: Combinatorics of crystal graphs and Kostka–Foulkes polynomials for the root systems B n ,C n and D n . Eur. J. Comb. 27, 526–557 (2006)
Lusztig, G.: Singularities, character formulas, and a q-analog of weight multiplicities. Astérisque 101–102, 208–227 (1983)
Macdonald, I.-G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford Mathematical Monograph. Oxford University Press, New York (1995)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lecouvey, C. Stabilization of the Brylinski-Kostant filtration and limit of Lusztig q -analogues. J Algebr Comb 27, 451–477 (2008). https://doi.org/10.1007/s10801-007-0097-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10801-007-0097-9