A Cartan decomposition for non-symmetric reductive spherical pairs of rank-one type and its application to visible actions (Q2290615)

Let \(G_{\mathbb C}\) be a connected complex semisimple Lie group and \(H_{\mathbb C}\) a complex closed subgroup of \(G_{\mathbb C}\). The pair \((G_{\mathbb C}, H_{\mathbb C})\) is called spherical, or the complex homogeneous space \(G_{\mathbb C}/H_{\mathbb C}\) is spherical, if a Borel subgroup of \(G_{\mathbb C}\) has an open orbit in \(G_{\mathbb C}/H_{\mathbb C}\). The classification of reductive spherical pairs \((G_{\mathbb C}, H_{\mathbb C})\), namely when \(H_{\mathbb C}\) is a reductive Lie group, has been given by \textit{M. Krämer} [Compos. Math. 38, 129--153 (1979; Zbl 0402.22006)] when \(G_{\mathbb C}\) is simple and by \textit{M. Brion} [Compos. Math. 63, 189--208 (1987; Zbl 0642.14011)], \textit{I. V. Mikityuk} [Mat. Sb., Nov. Ser. 129(171), No. 4, 514--534 (1986; Zbl 0621.70005)] respectively, when \(G_{\mathbb C}\) is semisimple. In the present paper the author provides new examples of a Cartan decomposition for non-symmetric reductive pairs, namely, reductive non-symmetric spherical pairs of rank one type. He also shows that the action of some compact group on a non-symmetric reductive spherical homogeneous space of rank-one type is strongly visible. For the entire collection see [Zbl 1426.22001].