Abstract: Classifying the irreducible unitary representations of a non-compact Lie group is one of the oldest open problems in representation theory. In these talks, I will discuss a new approach to this problem based on the Hodge theory of complex flag varieties.

In the first talk, I will explain the original conjecture of Wilfried Schmid and Kari Vilonen connecting Lie group representations to mixed Hodge modules on the flag variety and its recent proof by Vilonen and myself. The upshot is that unitarity of a representation can be read off from a canonical filtration, the Hodge filtration, defined geometrically via Beilinson-Bernstein localisation.

In the second talk, I will explain two applications to concrete unitarity questions, both joint work with Lucas Mason-Brown. The first application is a simple proof of the FPP Conjecture, which provides a strong bound on the infinitesimal characters of unitary representations. The second application is to the unitarity of the so-called unipotent representations, which are expected to form building blocks for the entire unitary dual. The proof works by showing that the Hodge filtration for such representations has a particularly simple form: it is precisely the quantisation filtration predicted by the Orbit Method philosophy, which has been a guiding principle in the subject since the 1960s.