| Speaker: |
Kari Vilonen (University of Melbourne) |
| Title: |
Unitary representations of Lie groups and Hodge theory |
| Date (JST): |
Tue, May 13, 2025, 13:30 - 15:00 |
| Place: |
Seminar Room A |
| Abstract: |
The determination of the unitary dual of a Lie group is a longstanding problem in mathematics. The unitary dual consist of all irreducible unitary representations of a given Lie group. The question is important for the purposes of L^2 harmonic analysis such as the Langlands program. There have been many attempts to solve this problem and to put it in some general conceptual context. For example, in the 1960’s Kostand and Kirillov introduced the orbit method where the idea is to quantize co-adjoint orbits. In this talk I will explain a new point of view to this question. The idea is that representations carry a Hodge structure and the Hodge structure can be used to determine unitarity. The Hodge structure arises from the theory of mixed Hodge modules of Saito. Along the way we obtain vanishing results for mixed Hodge modules on flag manifolds and a version of Saito’s Kodaira vanishing theorem for twisted mixed Hodge modules. The proofs will depend on deformation and wall crossing arguments as well as the explicit determination of the associated graded of certain mixed Hodge modules. This is joint work with Dougal Davis establishing earlier conjectures made by Schmid and myself several years ago.
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