Unique ergodicity on compact homogeneous spaces

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