A conjugacy theorem for symmetric spaces
Let \(G_{\mathbb{C}}\) be a semisimple complex Lie group and \(G\subseteq G_{\mathbb{C}}\) a real form. Let \(t_1,\dots, t_n\) be representatives of the \(G\)-conjugacy classes of Cartan subalgebras of \(g\). Then the set \[ \bigcup^n_{i=1} GN_{G_{\mathbb{C}}}(t_i)G \] contains an open and dense subset of \(G_{\mathbb{C}}\). This was proved by \textit{R. Stanton} [Am. J. Math. 108, 1411-1424 (1986; Zbl 0626.43008)] using a result of Rothschild which says that two Cartan subalgebras in \(g\) are conjugate under \(G_{\mathbb{C}}\) if and only if they are conjugate under \(G\). In [\textit{J. Hilgert} and \textit{K.-H. Neeb}, Lie semigroups and their applications, Lect. Notes Math. 1552 (Berlin 1993; Zbl 0807.22001)] and [\textit{J. D. Lawson}, J. Lie Theory 4, 17-29 (1994; Zbl 0824.22012)] this double coset decomposition was a main tool for examining maximal subsemigroups of \(G_{\mathbb{C}}\) that contain \(G\) nonisolated. In [\textit{A. N. Neumann} and \textit{G. Ólafsson}, Minimal and maximal subgroups related to causal symmetric spaces (submitted)] this decomposition was generalized to noncompactly causal semisimple symmetric spaces \(G/H\). It was shown there, that \[ \bigcup^n_{i=1} HN_G(a_i)H \] contains an open and dense subset of \(G\). Here the \(a_1,\dots, a_n\) are the representatives of \(H_0\)-conjugacy classes of Cartan subspaces in \(q\), the \((-1)\)-eigenspace of the involution in \(g\), where \(H_0\) is the connected component of \(H\) containing the identity. The essential fact used in the proof is that in noncompactly causal symmetric spaces the \(G\)-conjugacy classes of Cartan subspaces are the same as the \(H_0\)-conjugacy classes. In [\textit{A. Neumann} and \textit{G. Ólafsson} (op. cit.)] this was established by classifying the \(H_0\)-conjugacy classes and required detailed information on the special structure theory of noncompactly causal spaces. In this paper, we prove for general connected semisimple symmetric spaces \(G/H\) that two Cartan subspaces are conjugate under \(G\) if and only if they are conjugate under \(H\). As a consequence of our theorem, we can generalize the double coset decomposition to all semisimple symmetric spaces.
scientific article; zbMATH DE number 1336996