Invariant Measures and Orbit Closures on Homogeneous Spaces for Actions of Subgroups Generated by Unipotent Elements

The theorems of M. Ratner, describing the finite ergodic invariant measures and the orbit closures for unipotent flows on homogeneous spaces of Lie groups, are extended for actions of subgroups generated by unipotent elements. More precisely: Let G be a Lie group (not necessarily connected) and Γ a closed subgroup of G. Let W be a subgroup of G such that AdG(W ) is contained in the Zariski closure (in Aut(Lie G)) of the subgroup generated by the unipotent elements of AdG(W ). Then any finite W -invariant W -ergodic measure on G/Γ is a homogeneous measure (i.e., it is supported on a closed orbit of a subgroup preserving the measure). Moreover, if G/Γ has finite volume (i.e., has a finite G-invariant measure), then the closure of any orbit of W on G/Γ is a homogeneous set (i.e., a finite volume closed orbit of a subgroup containing W ). Both the above results hold if W is replaced by any subgroup Λ ⊂ W such that W/Λ has finite volume.