Limits of weight spaces, Lusztig’s \(q\)-analogs, and fiberings of adjoint orbits. (English) Zbl 0729.17005

Brylinski, Ranee Kathryn

J. Am. Math. Soc. 2, No. 3, 517-533 (1989).

Let \({\mathfrak g}\) be a compact semisimple Lie algebra with Cartan subalgebra \({\mathfrak h}\) and suppose \(V\) is a representation in a corresponding \({\mathcal O}\)-category. For each principal nilpotent \(e\in {\mathfrak g}\) compatible with \({\mathfrak h}\), the author constructs filtrations on the weight spaces of V. The Poincaré series of the associated graded modules are called the jump polynomials for \(V\). The main result in this paper says that if \(V=V_{\lambda}\) is finite dimensional (with highest weight \(\lambda\)) and \(\mu\) is a dominant weight, then the corresponding jump polynomial equals Lusztig’s \(q\)-analog, \(m^{\mu}_{\lambda}(q)\) of the weight multiplicity function [see G. Lusztig, Astérisque 101/102, 208–229 (1983; Zbl 0561.22013)].
Remark: The author’s assumption that either \({\mathfrak g}\) is classical or \(\mu\) is regular, can be omitted thanks to a recent result by B. Broer [“On the subregular nilpotent variety, Preprint No. 91-06, May 1991, University of Amsterdam].

Reviewer: H.H.Andersen (Aarhus)

MSC:

17B08	Coadjoint orbits; nilpotent varieties
17B10	Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
20G05	Representation theory for linear algebraic groups

Keywords:

principal nilpotent orbit; Poincaré series; jump polynomials; weight multiplicity