On semisimple classes and semisimple characters in finite reductive groups

Olivier Brunat[1]

  • [1] Université Denis Diderot - Paris 7 UFR de Mathématiques 175, rue du Chevaleret F-75013 Paris.

Annales de l’institut Fourier (2012)

  • Volume: 62, Issue: 5, page 1671-1716
  • ISSN: 0373-0956

Abstract

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In this article, we study the elements with disconnected centralizer in the Brauer complex associated to a simple algebraic group defined over a finite field with corresponding Frobenius map and derive the number of -stable semisimple classes of with disconnected centralizer when the order of the fundamental group has prime order. We also discuss extendibility of semisimple characters of the fixed point subgroup to their inertia group in the full automorphism group. As a consequence, we prove that “twisted” and “untwisted” simple groups of type are “good” in defining characteristic, which is a contribution to the general program initialized by Isaacs, Malle and Navarro to prove the McKay Conjecture in representation theory of finite groups.

How to cite

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Brunat, Olivier. "On semisimple classes and semisimple characters in finite reductive groups." Annales de l’institut Fourier 62.5 (2012): 1671-1716. <http://eudml.org/doc/251149>.

@article{Brunat2012,
abstract = {In this article, we study the elements with disconnected centralizer in the Brauer complex associated to a simple algebraic group $\mathbf\{G\}$ defined over a finite field with corresponding Frobenius map $F$ and derive the number of $F$-stable semisimple classes of $\mathbf\{G\}$ with disconnected centralizer when the order of the fundamental group has prime order. We also discuss extendibility of semisimple characters of the fixed point subgroup $\mathbf\{G\}^F$ to their inertia group in the full automorphism group. As a consequence, we prove that “twisted” and “untwisted” simple groups of type $E_6$ are “good” in defining characteristic, which is a contribution to the general program initialized by Isaacs, Malle and Navarro to prove the McKay Conjecture in representation theory of finite groups.},
affiliation = {Université Denis Diderot - Paris 7 UFR de Mathématiques 175, rue du Chevaleret F-75013 Paris.},
author = {Brunat, Olivier},
journal = {Annales de l’institut Fourier},
keywords = {algebraic groups; semisimple classes; Brauer complex; semisimple characters; finite reductive groups; disconnected centralizers; inductive McKay condition; finite algebraic groups; Brauer complexes; McKay conjecture; Gel'fand-Graev characters; numbers of irreducible characters},
language = {eng},
number = {5},
pages = {1671-1716},
publisher = {Association des Annales de l’institut Fourier},
title = {On semisimple classes and semisimple characters in finite reductive groups},
url = {http://eudml.org/doc/251149},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Brunat, Olivier
TI - On semisimple classes and semisimple characters in finite reductive groups
JO - Annales de l’institut Fourier
PY - 2012
PB - Association des Annales de l’institut Fourier
VL - 62
IS - 5
SP - 1671
EP - 1716
AB - In this article, we study the elements with disconnected centralizer in the Brauer complex associated to a simple algebraic group $\mathbf{G}$ defined over a finite field with corresponding Frobenius map $F$ and derive the number of $F$-stable semisimple classes of $\mathbf{G}$ with disconnected centralizer when the order of the fundamental group has prime order. We also discuss extendibility of semisimple characters of the fixed point subgroup $\mathbf{G}^F$ to their inertia group in the full automorphism group. As a consequence, we prove that “twisted” and “untwisted” simple groups of type $E_6$ are “good” in defining characteristic, which is a contribution to the general program initialized by Isaacs, Malle and Navarro to prove the McKay Conjecture in representation theory of finite groups.
LA - eng
KW - algebraic groups; semisimple classes; Brauer complex; semisimple characters; finite reductive groups; disconnected centralizers; inductive McKay condition; finite algebraic groups; Brauer complexes; McKay conjecture; Gel'fand-Graev characters; numbers of irreducible characters
UR - http://eudml.org/doc/251149
ER -

References

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