Topology of Hitchin systems and Hodge theory of character varieties

Given a compact Riemann surface of genus at least two, there are two algebraic varieties attached to it: the character variety Ch, and the Hitchin moduli space M. The non-Abelian Hodge theorem asserts that they are diffeomorphic (but have different complex structures). While the rational cohomology rings H*(Ch) and H*(M) are isomorphic, the mixed Hodge structures are different and so are the weight filtrations, which therefore cannot possibly correspond via the non Abelian Hodge theorem. In recent joint work with T. Hausel (Oxford) and L. Migliorini (Bologna) it is shown that the non-Abelian Hodge theorem exchanges the weight filtration on H*(Ch) with the (perverse) Leray filtration on H*(M) for the Hitchin map h on M. Moreover, curious symmetries, observed on H^*(Ch) by number-theoretic means, turn out to be the more familiar Lefschetz and Poincaré symmetries for the map h. (The perverse Leray filtration is formally analogous to the Leray filtration associated with the Leray spectral sequence. However, as my on-line thesaurus states: "perverse = resistant to guidance or discipline.")