On Hodge polynomials of Singular Character Varieties

Let $\mathcal{X}_{\Gamma}G:=\mathrm{Hom}(\Gamma,G)/\!/G$ be the $G$-character variety of $\Gamma$, where $G$ is a complex reductive group and $\Gamma$ a finitely presented group. We introduce new techniques for computing Hodge-Deligne and Serre polynomials of $\mathcal{X}_{\Gamma}G$, and present some applications, focusing on the cases when $\Gamma$ is a free or free abelian group. Detailed constructions and proofs of the main results will appear elsewhere.