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 G defined over a finite field with corresponding Frobenius map F and derive the number of F -stable semisimple classes of 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 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.

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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