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

TL;DR

This paper extends Ratner's theorems on invariant measures and orbit closures for unipotent flows to actions of subgroups generated by unipotent elements on homogeneous spaces, establishing conditions for measures and orbit closures to be homogeneous.

Contribution

It generalizes Ratner's results to broader subgroup actions, characterizing invariant measures and orbit closures as homogeneous in this setting.

Findings

Invariant measures are supported on homogeneous sets.

Orbit closures of subgroup actions are homogeneous when finite volume conditions are met.

Results apply to subgroups with finite volume quotients.

Abstract

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 Gamma a closed subgroup of G. Let W be a subgroup of G such that Ad(W) is contained in the Zariski closure (in the group of automorphisms of the Lie algebra of G) of the subgroup generated by the unipotent elements of Ad(W). Then any finite ergodic invariant measure for the action of W on G/Gamma is a homogeneous measure (i.e., it is supported on a closed orbit of a subgroup preserving the measure). Moreover, if G/Gamma has finite volume (i.e., has a finite G-invariant measure), then the closure of any orbit of W on G/Gamma 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 Lambda of W such that W/Lambda has finite volume.