  We discuss some applications of signature quantization to the representation
theory of compact Lie groups. In particular, we prove signature analogues of
the Kostant formula for weight multiplicities and the Steinberg formula for
tensor product multiplicities. Using symmetric functions, we also find, for
type A, analogues of the Weyl branching rule and the Gelfand-Tsetlin theorem.
These analogues involve a q-analogue of the Kostant partition function. We show
that in type A, this q-analogue is polynomial in the relative interior of the
cells of a complex of cones. This chamber complex can be taken to be the same
as the chamber complex of the usual Kostant partition function. We present the
case of A_2 as a detailed example.
