  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.
