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On the Brylinski-Kostant filtration. (English) Zbl 0991.17006
Let $${\mathfrak g}$$ be a split semisimple Lie algebra over a field of characteristic zero. The authors of the paper under review study a certain filtration (the so-called Brylinski-Kostant filtration) on a given weight space $$V_\mu$$ of a simple $${\mathfrak g}$$-module $$V$$. The $$n$$th subspace of this filtration consists of those vectors in $$V_\mu$$ that are annihilated by the $$n$$th power of a fixed regular nilpotent element in $${\mathfrak g}$$. Using a geometric approach R. K. Brylinski proved in [J. Am. Math. Soc. 2, 517-533 (1989; Zbl 0729.17005)] that for a dominant weight $$\mu$$ the Poincaré polynomial (in the variable $$q$$) associated to this filtration coincides with the coefficient of $$e^\mu$$ in a $$q$$-version of the ordinary character formula of $$V$$ introduced by G. Lusztig [Astérisque 101/102, 208-227 (1983; Zbl 0561.22013)]. The authors give a new proof of this result using algebraic and representation theoretic techniques and extend it to non-dominant weights as conjectured by the third author in his thesis.

##### MSC:
 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 17B20 Simple, semisimple, reductive (super)algebras 17B35 Universal enveloping (super)algebras
##### MathOverflow Questions:
Lusztig's $q$-analog of weight multiplicity with product formula
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##### References:
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