Calogero-Moser spaces vs unipotent representations

Cédric Bonnafé (1)

1 IMAG - Institut Montpelliérain Alexander Grothendieck

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 ${\mathbb{B}}^\times$-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 survey is to gather all these observations, state precise conjectures and provide general facts and examples supporting these conjectures.

Domaines: Théorie des représentations [math.RT]; Géométrie algébrique [math.AG]

Cédric Bonnafé. Calogero-Moser spaces vs unipotent representations. Pure and Applied Mathematics Quarterly, In press. ⟨hal-03565126v3⟩