On Lusztig's $q$-analogues of all weight multiplicities of a representation

Abstract

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. We obtain new properties of the $q$-analogue of weight multiplicities in finite-dimensional representations of $\mathfrak{g}$. In particular, it is proved that certain weighted sum of $q$-analogues of all weights of a representation $V$ equals the $q$-analogue of the zero weight multiplicity in the reducible representation $V\otimes V^{\ast }$. This also provides another formula for the $\mathbb{Z}[q]$-valued symmetric bilinear form on the character ring of $\mathfrak{g}$ that was introduced by R.Gupta (Brylinski) in 1987.

arXiv comment15 pages