Unitary representations of Lie groups and Hodge theory

The determination of the unitary dual of a Lie group is a longstanding problem. In this talk I will explain how the unitarity of a representation of a real reductive group can be read off from its Hodge filtration establishing a conjecture made by Wilfried Schmid and myself a while back. This is part of a larger set of conjectures which postulate that global sections of mixed Hodge modules on flag manifolds give rise to mixed Hodge structures. In joint work with Dougal Davis we have proved a substantial part of these conjectures and in particular the Hodge theoretic unitarity criterion. If time permits I will also discuss some general results about mixed Hodge modules that needed to be established on the way.