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/Λ is G-invariant, where Λ 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/Λ 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/Λ, and λ and μ denote the Haar probability measures on the homogeneous spaces Ux and Gx, respectively (cf. Ratner's theorem). Then aⁿλ → μ weakly as n → ∞. We also give an algebraic characterization of algebraic subgroups F < SL_n(ℝ) for which all orbit closures on SL_n(ℝ)/SL_n(ℤ) are finite-volume almost homogeneous, namely the smallest observable subgroup of SL_n(ℝ) containing F should have no non-trivial algebraic characters defined over ℝ.