On unipotent blocks and their ordinary characters
type: Item
instance of: scholarly article
title: On unipotent blocks and their ordinary characters
zbMATH Open document ID: 0817.20046
DOI: 10.1007/BF01232237
author: Marc Cabanes
author: Michel Enguehard
published in: Inventiones Mathematicae
publication date: 1995-07-31T00:00:00Z
full work available at URL: https://eudml.org/doc/144211
review text: Let \(G\) be a connected, reductive algebraic group defined over \(F_ q\), \(F : G \to G\) a Frobenius morphism and \(G^ F\) the finite group of \(F\)- fixed points of \(G\). Let \(\ell\) be an odd prime not dividing \(q\). Assuming that \(\ell\) is a good prime for \(G\) and \(\ell \neq 3\) if \(^ 3 D_ 4\) is involved in \(G\), the authors describe the unipotent \(\ell\)- blocks of \(G\), i.e. the \(\ell\)-blocks containing unipotent characters. Let \(L\) be an \(F\)-stable Levi subgroup of \(G\) and \(d\) a positive integer. Then \(L\) is said to be \(d\)-split if it is the centralizer of a \(\phi_ d\)-subgroup of \(G\) in the sense of \textit{M. Broué} and \textit{G. Malle} [Math. Ann. 292, 241-262 (1992)], where \(\phi_ d(x)\) is the \(d\)th cyclotomic polynomial. Let \(R^ G_ L\) and \(^*R^ G_ L\) denote the Lusztig functor and its adjoint. A unipotent \(d\)-pair is a pair \((L,\lambda)\) where \(L\) is a \(d\)-split Levi subgroup of \(G\) and \(\lambda\) is a unipotent character of \(L^ F\), and \((L,\lambda)\) is \(d\)-cuspidal if for every proper \(d\)-split Levi subgroup \(M\) of \(L\), \(^*R^ L_ M(\lambda) = 0\). The main theorem (4.4) proved by the authors then states that there is a bijection between \(G^ F\)-conjugacy classes of unipotent \(e\)-cuspidal pairs, where \(e\) is the order of \(q \bmod \ell\), and the set of unipotent \(\ell\)-blocks of \(G^ F\). If a block corresponds to a pair \((L,\lambda)\), the unipotent characters in the block are precisely the constituents of \(R^ G_ L(\lambda)\). The defect group of the block is a Sylow \(\ell\)-subgroup of \(C^ 0_ G([L,L])^ F\). Furthermore, all the characters in the block can be described in terms of Lusztig maps \(R^ G_{G(t)}\) where \(t\) is an \(\ell\)-element in the dual group \(G^{*F}\), and \(G(t) \subset G\) is in duality with \(C^ 0_{G*}(t)\). The proof involves an analysis of the centralizers of \(\ell\)-subgroups and \(e\)-split Levi subgroups, and local block theory. Remark: The theorem w…
reviewed by: Bhama Srinivasan
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 599649
zbMATH Keywords: blocks containing unipotent characters
zbMATH Keywords: conjugacy classes of unipotent cuspidal pairs
zbMATH Keywords: split Levi subgroups
zbMATH Keywords: connected reductive algebraic groups
zbMATH Keywords: Frobenius morphisms
zbMATH Keywords: Lusztig functors
zbMATH Keywords: unipotent \(\ell\)-blocks
zbMATH Keywords: defect groups
zbMATH Keywords: Lusztig maps
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 599649