The variety parametrizing twisted representations with values in a reductive group G of the fundamental group of a Riemann surface has a natural structure of affine smooth variety,   and, as such, has a mixed Hodge structure, recently investigated by   Hausel and Rodriguez-Villegas.
By Hitchin theory, this variety is   diffeomorphic to the moduli space of semistable principal bundles with a Higgs field.
This moduli space has a natural proper map, the Hitchin fibration, with target a linear space. We conjecture that the weight filtration of the cohomology of the character variety coincides with a filtration naturally associated to the topology of the Hitchin fibration.
We will discuss some evidence for this conjecture and a strategy.
Joint work with M. de Cataldo(SUNY) and T.Hausel(OXFORD)