A conjectural characterization of the weight filtration of the character variety of a Riemann surface

The (twisted) character variety of a Riemann surface, parametrizing the representations of its fundamental group into an algebraic group, has a natural structure of affine variety, and as such, it has a mixed Hodge structure, recently investigated, in the case the group is of type A, by Hausel and Rodriguez Villegas. I will discuss a conjectural characterization of the weight filtration in terms of a filtration of topological nature associated to the Hitchin map, and the evidence available for the conjecture. Work in progress with M.de Cataldo and T.Hausel.