Commuting involutions of semisimple groups (Q1192429)

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.