Cédric Bonnafé

Quasi-Isolated Elements in Reductive Groups

Abstract A semisimple element s of a connected reductive group G is called quasi-isolated (respectively isolated) if C G ( s ) (respectively ( s )) is not contained in a Levi subgroup of a proper parabolic subgroup of G . We study properties of quasi-isolated semisimple elements and give a classification in terms of the affine Dynkin diagram of G . Tables are provided for adjoint simple groups.

Semidirect Product Decomposition of Coxeter Groups

Let (W, S) be a Coxeter system, let S = I ∪ J be a partition of S such that no element of I is conjugate to an element of J, let be the set of W I -conjugates of elements of J, and let be the subgroup of W generated by . We show that and that is the canonical set of Coxeter generators of the reflection subgroup of W. We also provide algebraic and geometric conditions for an external semidirect product of Coxeter groups to arise in this way, and explicitly describe all such decompositions of (irreducible) finite Coxeter groups and affine Weyl groups.

Twisted invariant theory for reflection groups

Let G be a finite reflection group acting in a complex vector space V = ℂ r , whose coordinate ring will be denoted by S . Any element γ ∈ GL( V ) which normalises G acts on the ring S G of G -invariants. We attach invariants of the coset Gγ to this action, and show that if G ′ is a parabolic subgroup of G , also normalised by γ, the invariants attaching to G ′ γ are essentially the same as those of Gγ . Four applications are given. First, we give a generalisation of a result of Springer-Stembridge which relates the module structures of the coinvariant algebras of G and G ′ and secondly, we give a general criterion for an element of Gγ to be regular (in Springer’s sense) in invariant-theoretic terms, and use it to prove that up to a central element, all reflection cosets contain a regular element. Third, we prove the existence in any well-generated group, of analogues of Coxeter elements of the real reflection groups. Finally, we apply the analysis to quotients of G which are themselves reflection groups.

Elements unipotents reguliers des sous-groupes de Levi

Resume. Nous etudions la structure du centralisateur d’un element unipotent regulier d’un sousgroupe de Levi d’un groupe reductif, ainsi que la structure du groupe des composantes de ce centralisateur en relation avec la notion de systeme local cuspidal definie par Lusztig. Nous determinons son radical unipotent, montrons l’existence d’un complement de Levi et etudions la structure de son groupe de Weyl. Comme application, nous demontrons des resultats qui etaient annonces dans un precedent article de l’auteur sur les elements unipotents reguliers.

Actions of relative Weyl groups II

Abstract The aim of this second part is to compute explicitly the Lusztig restriction of the characteristic function of a regular unipotent class of a finite reductive group, improving slightly a theorem of Digne, Lehrer and Michel. We follow their proof but introduce new information, namely the computation of morphisms defined in the first part when v is a regular unipotent element. This new information is obtained by studying generalizations of the variety of companion matrices which were introduced by Steinberg.

On the endomorphism algebras of modular Gelfand–Graev representations

We study the endomorphism algebras of a modular Gelfand-Graev representation of a finite reductive group by investigating modular properties of homomorphisms constructed by Curtis and Curtis-Shoji.

Hecke algebras with unequal parameters and Vogan’s left cell invariants

In 1979, Vogan introduced a generalised τ-invariant for characterising primitive ideals in enveloping algebras. Via a known dictionary this translates to an invariant of left cells in the sense of Kazhdan and Lusztig. Although it is not a complete invariant, it is extremely useful in describing left cells. Here, we propose a general framework for defining such invariants which also applies to Hecke algebras with unequal parameters.

Regular unipotent elements

Abstract We construct a map from the set of regular unipotent classes of a finite reductive group to the corresponding set in a Levi subgroup. A theorem of Digne, Lehrer and Michel on Lusztig restriction of characteristic functions of such classes gives in particular another (not explicit) construction of such a map. We conjecture that these two maps coincide.

Free Coxeter Groups

This chapter, mostly taken from an article of Shi and Yang, contains the full description of cells and the proof of Lusztig’s Conjectures for free Coxeter groups (often called universal Coxeter groups).

Descent Sets, Knuth Relations and Vogan Classes

This chapter is devoted to applications of the results of the previous Chapter 8, where we investigated the general problem of relating the (left or right) cells of W and the (left or right) cells of all its standard parabolic subgroups. Thanks to parabolic subgroups of rank 2, we recall how Knuth relations are defined. We also explain how cellular maps can be used to construct an extension of Vogan’s generalized ?-invariant.