Skip to Main content Skip to Navigation
Theses

Les descentes de Shintani des groupes de Suzuki et de Ree

Abstract : The thesis is about representation theory of finite reductive groups. Such a group is defined as GF, where G is a reductive connected group over an algebraically closed field of characteristic p>0, and F is an endomorphism such that the fixed point subgroup GF is finite.
In this situation, we obtain a family of finite groups, when we replace F by Fm (m is a non-negative integer). This idea isimportant in the general theory of finite reductive groups, as developed by Lusztig. In the case where m=2, F acts as an automorphism on GF2. We can consider the semidirect product of GF2 by F. Then the "Shintani descent" defines an isometry between the space of class functions on the class GF2.F and the class function space of GF. The aim of the thesis is to study this isometry in the case where G is the Suzuki group or the Ree group of type G2, defined by a "very twisted" map F (in the sense that F is not a Frobenius map). The "very twisted" nature of F is the source of a number of problems as far as the general theory is concerned. We thus explicitly compute the table of the values of the class function on the set GF2.F.
As applications, we can explicitly determine the eigenvalues associed by Lusztig to the cuspidal unipotent characters of the Suzuki and the Ree group, and the Fourier matrices of these groups. We can verify many conjectures in the modular representation theory: Broue's conjecture, existence of basic set. More generally, the determination of the character tables of cyclic extensions is in the project to compute the character table of all extensions of finite simple groups.
Document type :
Theses
Complete list of metadatas

https://tel.archives-ouvertes.fr/tel-00012054
Contributor : Olivier Brunat <>
Submitted on : Tuesday, March 28, 2006 - 4:55:01 PM
Last modification on : Wednesday, July 8, 2020 - 12:43:14 PM
Long-term archiving on: : Saturday, April 3, 2010 - 10:08:11 PM

Identifiers

  • HAL Id : tel-00012054, version 1

Citation

Olivier Brunat. Les descentes de Shintani des groupes de Suzuki et de Ree. Mathématiques [math]. Université Claude Bernard - Lyon I, 2005. Français. ⟨tel-00012054⟩

Share

Metrics

Record views

372

Files downloads

248