Rational tori, semisimple orbits and the topology of hyperplane complements
type: Item
instance of: scholarly article
title: Rational tori, semisimple orbits and the topology of hyperplane complements
zbMATH Open document ID: 0816.20037
DOI: 10.1007/BF02566498
published in: Commentarii Mathematici Helvetici
publication date: 1993-01-16T00:00:00Z
full work available at URL: https://eudml.org/doc/140267
review text: Let \(G\) be a connected reductive group defined over the finite field \(\mathbb{F}_ q\) of \(q\) elements. Let \(F\) be the associated Frobenius endomorphism of \(G\). The author shows how to count regular semisimple classes in \(G^ F\) (resp. orbits in the Lie algebra \({\mathfrak g}^ F\)). If \(G\) is semisimple, he shows that the number of regular semisimple classes of \(G^ F\) is always odd, regardless of the characteristic, and the number of unipotent (resp. nilpotent) elements in \(G^ F\) (resp. \({\mathfrak g}^ F\)) is equal to the number of \(F\)-stable maximal tori of \(G^ F\), which is equal to \(q^{2N}\), where \(N\) is the number of positive roots of \(G\). Moreover, he shows that if \(\varepsilon\) is the alternating character of \(W\), the Weyl group of \(G\), and \(M_ W\) is the complexified hyperplane complement corresponding to \(W\), then \(\varepsilon\) does not occur in the cohomology modules \(H^*(M_ W)\), which generalises Stanley's result.
reviewed by: Ye Jiachen
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 17B20
zbMATH DE Number: 93806
zbMATH Keywords: unipotent elements
zbMATH Keywords: nilpotent elements
zbMATH Keywords: connected reductive groups
zbMATH Keywords: Frobenius endomorphism
zbMATH Keywords: number of regular semisimple classes
zbMATH Keywords: maximal tori
zbMATH Keywords: alternating character
zbMATH Keywords: Weyl groups
zbMATH Keywords: hyperplane complements
zbMATH Keywords: cohomology modules
author: Gus I. Lehrer
MaRDI profile type: Publication
OpenAlex ID: W1980194420
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description: scientific article; zbMATH DE number 93806