On actions of epimorphic subgroups on homogeneous spaces

Abstract

For an inclusion $F < G < L$ of connected real algebraic groups such that $F$ is epimorphic in $G$, we show that any closed $F$-invariant subset of $L/\Lambda$ is $G$-invariant, where $\Lambda$ is a lattice in $L$. This is a topological analogue of a result due to S. Mozes, that any finite $F$-invariant measure on $L/\Lambda$ is $G$-invariant.This result is established by proving the following result. If in addition $G$ is generated by unipotent elements, then there exists $a\in F$ such that the following holds. Let $U\subset F$ be the subgroup generated by all unipotent elements of $F$, $x\in L/\Lambda$, and $\lambda$ and $\mu$ denote the Haar probability measures on the homogeneous spaces $\overline{Ux}$ and $\overline{Gx}$, respectively (cf. Ratner's theorem). Then $a^n\lambda\to\mu$ weakly as $n\to\infty$.We also give an algebraic characterization of algebraic subgroups $F<{\rm SL}_n(\mathbb{R})$ for which all orbit closures on ${\rm SL}_n(\mathbb{R})/{\rm SL}_n(\Z)$ are finite-volume almost homogeneous, namely the smallest observable subgroup of ${\rm SL}_n(\mathbb{R})$ containing $F$ should have no non-trivial algebraic characters defined over $\mathbb{R}$.