Abstract
Let $X_\GammaG:=Hom(\Gamma,G)/\!/G$ 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 $X_\GammaG$, 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.
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