Unique ergodicity on compact homogeneous spaces

Abstract

Extending results of a number of authors, we prove that if U U is the unipotent radical of an R \mathbb {R} -split solvable epimorphic subgroup of a real algebraic group G G which is generated by unipotents, then the action of U U on G / Γ G/\Gamma is uniquely ergodic for every cocompact lattice Γ \Gamma in G G . This gives examples of uniquely ergodic and minimal two-dimensional flows on homogeneous spaces of arbitrarily high dimension. Our main tools are the Ratner classification of ergodic invariant measures for the action of a unipotent subgroup on a homogeneous space, and a simple lemma (the ‘Cone Lemma’) about representations of epimorphic subgroups.