  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.
