Unique ergodicity on compact homogeneous spaces

Abstract:

Extending results of a number of authors, we prove that if $U$ is the unipotent radical of an $\mathbb {R}$-split solvable epimorphic subgroup of a real algebraic group $G$ which is generated by unipotents, then the action of $U$ on $G/\Gamma$ is uniquely ergodic for every cocompact lattice $\Gamma$ in $G$. 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.