Unitary representations of real groups and localisation theory for Hodge modules

Abstract: I will explain recent joint work with Kari Vilonen, in which we prove a conjecture of Schmid and Vilonen linking mixed Hodge modules on flag varieties to unitary representations of real reductive Lie groups. The main idea behind our work is to upgrade Beilinson-Bernstein localisation from D-modules to mixed Hodge modules. When it applies, this endows everything in sight with a canonical filtration, the Hodge filtration, which we prove has some extremely nice properties, such as cohomology vanishing and global generation. In the context of real groups, we also prove that the Hodge filtration “sees” exactly which representations are unitary. We hope that this will lead to new progress on the very old problem of determining the unitary dual of a real group.

algebraic geometryrepresentation theory

Audience: researchers in the topic

Algebra and Geometry Seminar @ HKUST

Series comments: Algebra and Geometry seminar at the Hong Kong University of Science and Technology (HKUST).

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