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

TL;DR

This paper explores new properties of Lusztig's $q$-analogues of weight multiplicities in finite-dimensional representations of complex semisimple Lie algebras, revealing relations to tensor products and bilinear forms.

Contribution

It establishes a novel relation between weighted sums of $q$-analogues and the $q$-analogue of zero weight multiplicity, providing new formulas for symmetric bilinear forms on the character ring.

Findings

Weighted sum of $q$-analogues equals the $q$-analogue of zero weight multiplicity in $V\otimes V^*$

Provides a new formula for the symmetric bilinear form on the character ring

Enhances understanding of $q$-analogues in representation theory

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 $V\otimes V^*$. 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.