A parametrization of unipotent representations

Abstract

0.1. Let G be a simple algebraic group defined and split over a finite field Fq. Let U be the set of isomorphism classes of irreducible unipotent representations (over C) of the finite group G(Fq). Let W be the Weyl group of G and let Irr(W ) be the set of isomorphism classes of irreducible representations (over C) of W . In [L79] a partition of Irr(W ) into families is described and in [L84] a partition U = ⊔cUc of U (with c running over the families of Irr(W )) is introduced. Moreover, in [L84, §4] to any family c we have associated a finite group Gc and a bijection