On blocks of finite reductive groups and twisted induction
type: Item
instance of: scholarly article
title: On blocks of finite reductive groups and twisted induction
zbMATH Open document ID: 0954.20023
author: Marc Cabanes
author: Michel Enguehard
published in: Advances in Mathematics
publication date: 2000-11-07T00:00:00Z
review text: Let \(\mathbf G\) be a connected reductive algebraic group defined over \(\mathbb{F}_q\) and \(F\colon\mathbf G\to\mathbf G\) a Frobenius morphism, so that \(G:=\mathbf G^F\) is the finite group of \(F\)-invariant points of \(\mathbf G\). Let \(\ell\) be a prime not dividing \(q\) and \(e\) the order of \(q\bmod\ell\). Let \(\mathbf L\) be an \(F\)-stable Levi subgroup of \(\mathbf G\). The Lusztig induction functor \(R^{\mathbf G}_{\mathbf L}\) takes generalized characters of \(L\) (over \(\overline{\mathbf Q}_\ell\)) into generalized characters of \(G\). The main purpose of the paper is to describe the \(\ell\)-blocks of \(G\) in terms of the functor \(R_{\mathbf L}^{\mathbf G}\). It has been proved in several cases that the \(\ell\)-blocks of \(G\) induce a partition of the set of irreducible ordinary characters of \(G\) similar to that of ordinary Harish-Chandra theory. Ordinary induction is replaced with Lusztig induction and the partition into series is given by generalized Harish-Chandra theory, first introduced by \textit{P.~Fong} and \textit{B.~Srinivasan} for unipotent characters [in J. Algebra 104, 301-309 (1986; Zbl 0606.20035)]. Rational Levi subgroups of rational parabolic subgroups are replaced by \(e\)-split Levi subgroups of \(\mathbf G\) (the order of \(G\) is a polynomial in \(q\), and one can talk of the ``polynomial order'' of the pair \((\mathbf G,F)\); the centralizer in \(\mathbf G\) of an \(F\)-stable torus whose polynomial order is a power of the \(e\)th cyclotomic polynomial is called an \(e\)-split Levi subgroup of \(\mathbf G\)), cuspidal irreducible characters are replaced by \(e\)-cuspidal irreducible characters (an irreducible character \(\chi\) of \(G\) is said to be \(e\)-cuspidal if \(^*R^{\mathbf G}_{\mathbf L}(\chi)=0\) for every proper \(e\)-split Levi subgroup \(\mathbf L\) of \(\mathbf G\), where \(^*R^{\mathbf G}_{\mathbf L}\) denotes the adjoint of the functor \(R^{\mathbf G}_{\mathbf L}\)). The \(\ell\)-blocks of \(G\) have…
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 1340110
zbMATH Keywords: theory of blocks
zbMATH Keywords: Lusztig induction functors
zbMATH Keywords: finite groups of Lie type
zbMATH Keywords: modular representations
zbMATH Keywords: connected reductive algebraic groups
zbMATH Keywords: generalized characters
zbMATH Keywords: generalized Harish-Chandra theory
zbMATH Keywords: Levi subgroups
zbMATH Keywords: cuspidal irreducible characters
reviewed by: Anne-Marie Aubert
MaRDI profile type: Publication
full work available at URL: aima.1998.1814
OpenAlex ID: W2030121927
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DOI: 10.1006/AIMA.1998.1814
description: scientific article; zbMATH DE number 1340110