On the self-dual representations of finite groups of Lie type
type: Item
instance of: scholarly article
title: On the self-dual representations of finite groups of Lie type
zbMATH Open document ID: 0922.20020
author: Dipendra Prasad
published in: Journal of Algebra
publication date: 1999-08-22T00:00:00Z
review text: The author presents various results on self-dual irreducible complex representations of a finite group \(G\) of Lie type, motivated in part by known related results in the theory of representations of semisimple algebraic groups. In the context of finite groups, the self-dual irreducible representations are those with real-valued character. The author concentrates on self-dual irreducible generic representations, which may be interpreted as constituents of the Gelfand-Graev representations of \(G\). Sufficient conditions are given in terms of the action of the centre of \(G\) for a self-dual irreducible generic representation of \(G\) to be of orthogonal type (equivalently, to be the character of a real representation of \(G\)). These conditions imitate the algebraic group theory. Perhaps the author's most novel result is the following. Let \(q\) be a power of a prime with \(q\equiv 3\bmod 4\). Then there exists an irreducible self-dual complex representation of the special linear group \(\text{SL}(6,q)\) which is of orthogonal type although its central involution acts non-trivially. Similarly, there exists an irreducible self-dual complex representation of the special linear group \(\text{SL}(6,q)\) which is of symplectic type although its central involution acts trivially (the representation may be viewed as faithful representation of the projective group \(\text{PSL}(6,q)\)). Such representations are uncommon and might be thought unlikely to exist (they do not exist in the case when \(q\equiv 1\bmod 4\)). However, it is now known that they also exist in \(\text{SL}(4n+2,q)\) when \(n\geq 1\) and \(q\equiv 3\bmod 4\).
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 1247579
zbMATH Keywords: finite groups of Lie type
zbMATH Keywords: self-dual irreducible complex representations
zbMATH Keywords: generic representations
zbMATH Keywords: special linear groups
Wikidata QID: Q115395863
reviewed by: Rod Gow
MaRDI profile type: Publication
full work available at URL: jabr.1998.7550
OpenAlex ID: W1965083241
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DOI: 10.1006/JABR.1998.7550
description: scientific article; zbMATH DE number 1247579