Calogero-Moser spaces vs unipotent representations

Geometric Representation Theory Seminar

Organizers：

Lin Chen, Will Donovan, Penghui Li, Peng Shan, Changjian Su, Wenbin Yan

Speaker：

Cédric Bonnafé (CNRS, Université Montpellier)

Time：

Wed., 15:30-16:30

Oct. 23, 2024

Venue：

Ning Zhai (新宁斋) 101

Title：

Calogero-Moser spaces vs unipotent representations

Abstract：

Lusztig's classification of unipotent representations of finite reductive groups depends only on the associated Weyl group W (endowed with its Frobenius automorphism). All the structural questions (families, Harish-Chandra series, partition into blocks...) have an answer in a combinatorics that can be entirely built directly from W . Over the years, we have noticed that the same combinatorics seems to be encoded in the Poisson geometry of a Calogero-Moser space associated with W (roughly speaking, families correspond to ${\mathbb{C}}^\times$-fixed points, Harish-Chandra series correspond to symplectic leaves, blocks correspond to symplectic leaves in the fixed point subvariety under the action of a root of unity). The aim of this talk is to gather these observations, state precise conjectures and provide general facts and examples supporting these conjectures.