Character varieties are spaces of representations of finitely presented groups into real reductive Lie groups. In some cases, these can be interpreted as moduli spaces of G-Higgs bundles over a Kähler manifold M, spaces which reflect both algebraic properties of G and complex analytic properties of M. In particular, their topology is a very important subject of research.

When G is a complex algebraic group, these spaces have more refined invariants, such as Deligne's mixed Hodge structures, which are typically very hard to calculate, especially in the singular setting. In this seminar, we present the explicit computation of the mixed Hodge polynomial of the G-character variety of a free abelian group of rank r, when G is (P)SL(n,C) or GL(n,C) for all values of r and n.

Key steps in the computation are equivariant versions of these polynomials under the symmetric groups, and Cheah's beautiful generating series for the mixed Hodge polynomials of symmetric products. Remarkably, (for even r) the polynomials exhibit the "curious Poincaré duality" first observed by Hausel--Rodriguez-Villegas in twisted character varieties of surface groups.
This is joint work with J. Silva.