Duality for representations of a reductive group over a finite field. II
type: Item
instance of: scholarly article
title: Duality for representations of a reductive group over a finite field. II
zbMATH Open document ID: 0535.20020
DOI: 10.1016/0021-8693(83)90202-8
author: Pierre Deligne
author: G. Lusztig
published in: Journal of Algebra
publication date: 1983-01-01T00:00:00Z
review text: The first part of this paper appeared in J. Algebra 74, 284-291 (1982; Zbl 0482.20027). The group is the set \(G\) of rational points of a connected reductive group \({\mathcal G}\) over a finite field \(F\); the duality operation has been defined for representations in the first part, from the homology of a complex associated to the fixed vectors of the unipotent radicals of parabolic subgroups defined over \(F\). This second part shows that for the virtual representations introduced by the authors [in Ann. Math., II. Ser. 103, 103-161 (1976; Zbl 0336.20029)] the duality operation is multiplication by a sign which is given. To prove this result, they use some orthogonality relations satisfied by a twisted induction from representations of subgroups of G coming from the Levi subgroups of \({\mathcal G}\) defined over \(F\); the article gives a proof of these orthogonality relations.
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 3849483
zbMATH Keywords: rational points
zbMATH Keywords: connected reductive group
zbMATH Keywords: unipotent radicals
zbMATH Keywords: parabolic subgroups
zbMATH Keywords: virtual representations
zbMATH Keywords: duality operation
zbMATH Keywords: orthogonality relations
zbMATH Keywords: twisted induction
zbMATH Keywords: Levi subgroups
MaRDI profile type: Publication
full work available at URL: 0021-8693(83)90202-8
OpenAlex ID: W2035423620
cites work: Representations of reductive groups over finite fields
cites work: Duality for representations of a reductive group over a finite field
cites work: On the finiteness of the number of unipotent classes
description: scientific article; zbMATH DE number 3849483