Hodge theory and the Langlands correspondence for real groups

The Langlands correspondence for real groups is a classification of irreducible admissible representations of a real reductive group in terms of Langlands parameters associated with the dual group. It was conjectured by Soergel (and proved in a restricted setting by Bezrukavnikov and Vilonen) that this categorifies to a Koszul duality between categories of representations and categories of perverse sheaves on spaces of Langlands parameters. In this talk, I will discuss work in progress with Yau Wing Li and Kari Vilonen, which aims to enhance this picture to include the Hodge filtrations on representations introduced by Schmid and Vilonen. I will state a conjecture (or “theorem in progress”) on a Langlands dual description of the category of filtered representations and the functor endowing a representation with its canonical Hodge filtration. As well as Hodge theory, our picture incorporates Soergel’s conjecture, Vogan’s theory of lowest K-types, and (when combined with unpublished results of Tsao-Hsien Chen, David Nadler and Lingfei Yi) implies a Betti geometric Langlands statement for the twistor P^1.