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Sur les caractères des groupes réductifs finis à centre non connexe: applications aux groupes spéciaux linéaires et unitaires
Cédric Bonnafé, Université de Franche-Comté, Besancon, France
A publication of the Société Mathématique de France.
cover
Astérisque
2006; 165 pp; softcover
Number: 306
ISBN-10: 2-85629-190-2
ISBN-13: 978-2-85629-190-0
List Price: US$52
Individual Members: US$46.80
Order Code: AST/306
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A first aim of this paper is to present an overview of results obtained by several authors on the characters of finite reductive groups with non-connected centre. The author is particularly interested in problems directly linked to the non-connectedness of the centre. He emphasises Gelfand-Graev and semisimple characters.

A second aim is to study the influence of the non-connectedness of the centre on the theory of character sheaves. The author studies more precisely the family of character sheaves whose support meets the regular unipotent class: these are analogues of the semisimple characters.

The last aim is the application of these results to finite reductive groups of type \(A\), split or not (as for instance the special linear or special unitary groups). Whenever the cardinality of the finite field is large enough, the author obtains a parametrization of the irreducible characters, a parametrization of the character sheaves, and he shows that the characteristic functions of character sheaves are Fourier transforms of the irreducible characters (Lusztig's conjecture). This gives a theoretical algorithm for computing the character table of these groups.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

Readership

Graduate students and research mathematicians interested in representation theory and linear algebraic groups over finite fields.

Table of Contents

  • Introduction
  • Préliminaires, notations, définitions
  • Le groupe \(\mathcal{Z}(\mathbf{G})\)
  • Induction et restriction de Lusztig, séries de Lusztig
  • Théorie de Harish-Chandra
  • Autour des caractères de Gelfand-Graev
  • Faisceaux-caractères
  • Groupes de type A
  • Le groupe spécial linéaire
  • A. Produits en couronne
  • B. Sommes de Gauss
  • Bibliographie
  • Index
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