On conjugacy classes of closed subgroups and stabilizers of Borel actions of Lie groups

Let G be a connected Lie group and C(G) be the space of all closed subgroups of G, equipped with the Fell topology. We consider actions on C(G), of G (by conjugation) and of groups of Lie automorphisms of G (by set operation). We describe conditions under which the orbit of a closed subgroup with finitely many connected components is locally closed (open in its closure) in C(G). The results are applied to conclude, in particular, that for any ergodic action of G on a standard Borel space the connected components of the identity in almost all the stabilizers are isomorphic to each other, and that if all the stabilizers are assumed to have only finitely many connected components then almost all of them are isomorphic to each other. Various conditions are also described under which almost all the stabilizers are conjugate to each other.