Mixed Hodge modules and real groups

Dougal Davis (University of Melbourne)

Wednesday 15th March, 2023 16:00-17:00 Maths 311B


I will explain an ongoing program, joint with Kari Vilonen, that aims to study unitary representations of real reductive Lie groups using mixed Hodge modules on flag varieties. The program revolves around a conjecture of Schmid and Vilonen that the signatures of Hermitian forms on representations are controlled by Hodge filtrations coming from geometry. This conjecture is expected to help determine the unitary irreducible representations by placing the problem in a more conceptual context. Our results to date centre around the interaction of Hodge theory with the unitarity algorithm of Adams, van Leeuwen, Trapa and Vogan, which calculates the signature of a canonical Hermitian form on an arbitrary representation by reducing to the case of tempered representations using deformations and wall crossing. The main consequences so far are a Hodge-theoretic proof of the ALTV wall crossing formula, and a proof of the Schmid-Vilonen conjecture for tempered representations.

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