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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
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