Twisted conjugacy in simply connected groups.
type: Item
instance of: scholarly article
title: Twisted conjugacy in simply connected groups.
zbMATH Open document ID: 1112.20041
author: Tonny A. Springer
published in: Transformation Groups
publication date: 2007-01-24T00:00:00Z
review text: \(G\) is a simply connected semisimple group over the algebraically closed field \(k\) and \(\sigma\) is a non-trivial diagram automorphism of \(G\). One has the twisted conjugation action of \(G\) on itself \(((g,h)\mapsto gh(\sigma g)^{-1})\). This paper collects a number of results concerning this action. Some samples: The fixed point group \(G^\sigma\) is semi-simple. The geometric quotient \(G//_\sigma G\) of \(G\) by the twisted action is isomorphic to \(\mathbb{A}^d\), where \(d=\dim T^\sigma\) for a \(\sigma\)-invariant maximal torus \(T\) in a \(\sigma\)-invariant Borel subgroup \(B\). The twisted conjugates \(g\) of elements of \(T\) form a dense subset of \(G\). They have a connected \(\sigma\)-centralizer \(Z_\sigma(g)=\{x\in G\mid xg(\sigma x)^{-1}=g\}\).
reviewed by: Wilberd van der Kallen
Mathematics Subject Classification ID: 20G15
zbMATH DE Number: 5118045
zbMATH Keywords: twisted conjugation actions
zbMATH Keywords: diagram automorphisms
zbMATH Keywords: simply connected semisimple groups
MaRDI profile type: Publication
full work available at URL: s00031-005-1113-6
OpenAlex ID: W2038695727
DOI: 10.1007/S00031-005-1113-6
description: scientific article; zbMATH DE number 5118045