Dougal Davis: D-modules workshop talk 3

Presenter: Dougal Davis, University of Melbourne
28 September 2023
Abstract: In these talks, we will explain some very recent results on mixed Hodge modules and the unitary dual of a real reductive Lie group. (With a little luck, the ink will have dried and there will be a preliminary version on the arXiv by the time the workshop starts.) The main idea behind our work is to upgrade Beilinson-Bernstein localisation from D-modules to mixed Hodge modules, following a proposal made by Schmid and Vilonen over 10 years ago. 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. We’ll do our best to put this problem in context, explain what our theorems say, and give the main ideas behind the proof.
D-modules webpage: sites.google.com/view/2023dmo...