On actions of epimorphic subgroups on homogeneous spaces

For an inclusion F < G < L of connected real algebraic groups such that F is epimorphic in G, we show that any closed F -invariant subset of L/3 is G-invariant, where 3 is a lattice in L. This is a topological analogue of a result due to S. Mozes, that any finite F -invariant measure on L/3 is G-invariant. This result is established by proving the following result. If in addition G is generated by unipotent elements, then there exists a ∈ F such that the following holds. Let U ⊂ F be the subgroup generated by all unipotent elements of F , x ∈ L/3, and λ and μ denote the Haar probability measures on the homogeneous spacesUx andGx, respectively (cf. Ratner’s theorem). Then anλ→ μ weakly as n→∞. We also give an algebraic characterization of algebraic subgroups F < SLn(R) for which all orbit closures on SLn(R)/SLn(Z) are finite-volume almost homogeneous, namely the smallest observable subgroup of SLn(R) containing F should have no nontrivial algebraic characters defined over R.