We prove a conjecture in \cite{ L} stating that certain polynomials $P^{ \sigma}_{ y,w}(q)$ 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 $P^{ \sigma}_{ y,w}(q)$.
Categories: math.RT