Unipotent representations and mixed Hodge modules

Let \Pi(G) denote the set of irreducible unitary representations of a semisimple Lie group G. A fundamental problem in representation theory is to describe the structure of this set. In previous joint work with Losev and Matvieievskyi, we have defined a class of representations called rigid unipotent representations, which are conjectured to form the building blocks of \Pi(G). Unfortunately, it is not at all clear from their construction that these representations are unitary. In 2011, Schmid and Vilonen proposed a geometric framework for studying unitary representations using Saito’s theory of mixed Hodge modules. In this talk, I will explain how this framework can be applied to prove the unitarity of all rigid unipotent representations. This is based on joint work in progress with Dougal Davis.