Actions of semisimple Lie groups and orbits of Cartan subgroups (Q1176769)

The author gives a new proof of the theorem of Shnol, based on the following idea. Let \(G\) be a real semisimple Lie group of rank \(r\), acting analytically and effectively on an \(n\)-dimensional (connected) real analytic manifold \(\mathcal M\), and \(H\) be a Cartan subgroup of \(G\). Then the existence of a point \(m\in{\mathcal M}\) is proved whose stabilizer \(H_ m\) in \(H\) is a discrete subgroup. Also, it is shown that each \(H\)- orbit is isotropic in the case of symplectic \(\mathcal M\) and Hamiltonian \(G\)-action. It is noticed that if \(G\) is compact, then the existence of a point whose stabilizer in \(H\) is trivial, is an immediate corollary of Mostov's finiteness theorem for conjugacy classes of stabilizers.