On the endomorphism algebra 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.Comment: 25 page

Two-sided cells in type BB (asymptotic case)

We compute two-sided cells of Weyl groups of type $B$ for the "asymptotic" choice of parameters. We also obtain some partial results concerning Kazhdan-Lusztig conjectures in this particular case.Comment: 20 pages, some misprints have been cleaned up in this second versio

Cells and cacti

Let $(W,S)$ be a Coxeter system, let $\varphi$ be a weight function on $S$ and let ${\mathrm{Cact}}\_W$ denote the associated {\it cactus group}. Following an idea of I. Losev, we construct an action of ${\mathrm{Cact}}\_W \times {\mathrm{Cact}}\_W$ on $W$ which has nice properties with respect to the partition of $W$ into left, right or two-sided cells (under some hypothesis, which hold for instance if $\varphi$ is constant or if $W$ is finite of rank \textless{} 5). It must be noticed that the action depends heavily on $\varphi$.Comment: 23 pages. This new version extends the scope of validity of the main result by removing an hypothesis. For this purpose, we have slightly extended some results of arXiv:1502.0166

Constructible characters and b-invariants

To each finite Coxeter system (W,S) and to each weight function L, Lusztig has defined the notions of constructible characters and of Lusztig families of W, using the so-called J-induction. Whenever L is constant, and using a general argument, Lusztig has shown that all Lusztig family contains a unique character with minimal b-invariant, and that every constructible character contains an irreducible constituent with minimal b-invariant. We show in this paper that this can be generalized to the case where L is not constant: our proof is by a case-by-case analysis.Comment: 12 page

Representation theory of Mantaci-Reutenauer algebras

We study some aspects of the representation theory of Mantaci-Reutenauer algebras: Cartan matrix, Loewy length, modular representations.Comment: 41 pages; the last question of the last section has been slightly modifie

A note on the Grothendieck ring of the symmetric group

Let $p$ be a prime number and let $n$ be a non-zero natural number. We compute the descending Loewy series of the algebra $R\_n/pR\_n$, where $R\_n$ denotes the ring of virtual ordinary characters of the symmetric group $S\_n$.Comment: 5 page

A progenerator for representations of SL(n,q) in transverse characteristic

Let G=GL(n,q), SL(n,q) or PGL(n,q) where q is a power of some prime number p, let U denote a Sylow p-subgroup of G and let R be a commutative ring in which p is invertible. Let D(U) denote the derived subgroup of U and let e be the central primitive idempotent of the group algebra RD(U) corresponding to the projection on the invariant RD(U)-submodule. The aim of this note is to prove that the R-algebras RG and eRGe are Morita equivalent (through the natural functor sending an RG-module M to the eRGe-module eM).Comment: 4 page

On the Calogero-Moser space associated with dihedral groups

Using the geometry of the associated Calogero-Moser space, R. Rouquier and the author have attached to any finite complex reflection group $W$ several notions (Calogero-Moser left, right or two-sided cells, Calogero-Moser cellular characters), completing the notion of Calogero-Moser families defined by Gordon. If moreover $W$ is a Coxeter group, they conjectured that these notions coincide with the analogous notions defined using the Hecke algebra by Kazhdan and Lusztig (or Lusztig in the unequal parameters case). In the present paper, we aim to investigate these conjectures whenever $W$ is a dihedral group.Comment: 28 page