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

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