A $(-q)$-analogue of weight multiplicities

We prove a conjecture in \cite{L} stating that certain polynomials $P_{y,w}^{\sigma }(q)$ introduced in \cite{LV1} for twisted involutions in an affine Weyl group give $(-q)$-analogues of weight multiplicities of the Langlands dual group $\check{G}$. We also prove that the signature of a naturally defined hermitian form on each irreducible representation of $\check{G}$ can be expressed in terms of these polynomials $P_{y,w}^{\sigma }(q)$.