Hodge theory and Langlands duality for real groups

Abstract: In this talk, I will outline a program (which is very much work in progress) which aims to give a Langlands dual description of Schmid and Vilonen’s Hodge theory for representations of real reductive Lie groups. The central point of Schmid-Vilonen theory is that irreducible representations carry canonical Hodge filtrations, which can be used to control analytic properties such as unitarity. The Langlands program, on the other hand, predicts that at least a large subset of the unitary representations can be parametrised using the dual reductive group. I will explain a network of conjectures (old and new) and theorems-in-progress that take the best of both worlds by describing both the category of representations and the formation of Hodge filtrations in terms of the dual group. Part of this is joint with Yau Wing Li and Kari Vilonen.