Repr\'esentations de Springer pour les groupes de r\'eflexions complexes imprimitifs

TL;DR

This paper introduces a finite set associated with spetsial complex reflection groups, providing a combinatorial parametrization for imprimitive groups that links Green functions and cyclotomic Hecke algebras.

Contribution

It offers a new combinatorial parametrization for certain complex reflection groups using Malle-Shoji symbols, connecting different areas of algebraic representation theory.

Findings

Parametrization of unipotent-like classes for imprimitive groups

Link between Green functions and cyclotomic Hecke algebras

Use of Malle-Shoji generalized symbols

Abstract

To a spetsial complex reflection group, equipped with a root lattice in the sense of Nebe, we attach a certain finite set playing a role which is analogous to the role of the set of unipotent classes of an algebraic group. In the case of imprimitive groups, we give a combinatoric parametrization of it in terms of Malle-Shoji generalized symbols. This result provides a link between the works of Shoji on Green functions for complex reflection groups and of Broue, Kim, Malle, Rouquier, et. al. on the cyclotomic Hecke algebras and their families of characters. ----- A un groupe de reflexions complexe spetsial, muni d'un reseau radiciel au sens de Nebe, nous associons un certain ensemble fini qui doit jouer un role analogue a celui de l'ensemble des classes unipotentes d'un groupe algebrique. Dans le cas des groupes imprimitifs, nous en donnons un parametrage combinatoire en termes des symboles generalises de Malle et Shoji. Ce resultat fournit un lien entre les travaux de Shoji sur les fonctions de Green pour les groupes de reflexions complexes et ceux de Broue, Kim, Malle, Rouquier, et al. sur les algebres de Hecke cyclotomiques et leurs familles de caracteres.