Commuting involutions of semisimple groups
type: Item
instance of: scholarly article
title: Commuting involutions of semisimple groups
zbMATH Open document ID: 0786.22019
author: Tadashi Nagano
author: Jiro Sekiguchi
published in: Tokyo Journal of Mathematics
publication date: 1992-09-27T00:00:00Z
review text: Let \(G\) be a connected semisimple Lie group, let \(\tau\) be a holomorphic involution of \(G\) and \(\theta\) a Cartan involution of \(G\) commuting with \(\tau\). Then the symmetric space \(G/G^ \tau\) is a vector bundle over the compact symmetric space \(M=G^ \theta/G^{\theta,\tau}\) (where superscripts denote the fixed points subgroups of \(\tau\), resp. \(\theta\), resp. \(\theta\) and \(\tau\)). To a closed geodesic through the origin \(o\) in \(M\), the authors associate a third involution \(\sigma\) of \(G\), arising from the symmetry with respect to the antipodal point of \(o\) on the geodesic. Then, \(\theta\), \(\sigma\), \(\tau\) commute to each other and (at the Lie algebra level) the symmetric pair \((G^{\theta\sigma}, G^{\theta\sigma,\tau})\) is of the so-called \(K_ \varepsilon\)-type studied by Oshima and Sekiguchi. All \(K_ \varepsilon\)-type pairs can be obtained in this way, whence a geometric characterization of those pairs. Under different assumptions on \(G\) and \(\tau\), the authors also obtain a generalization of the Borel embedding of a Hermitian symmetric space of the non-compact type into its compact dual.
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 17B40
zbMATH DE Number: 60843
zbMATH Keywords: commuting involutions
zbMATH Keywords: connected semisimple Lie group
zbMATH Keywords: symmetric space
zbMATH Keywords: symmetric pair
zbMATH Keywords: \(K_ \varepsilon\)-type pairs
zbMATH Keywords: Borel embedding
zbMATH Keywords: Hermitian symmetric space
reviewed by: François Rouvière
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270130375
OpenAlex ID: W2069622778
DOI: 10.3836/TJM/1270130375
description: scientific article; zbMATH DE number 60843