Institute for Mathematical Sciences

Preprint ims99-7b

N. Shah and B. Weiss
On Actions of Epimorphic Subgroups on Homogeneous Spaces

Abstract: We show that for an inclusion $F The key ingredient in establishing this result is the study of the limiting distributions of certain translates of a homogeneous measure. We show that 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 $\cl{Ux}$ and $\cl{Gx}$, 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 $F<\SL_n(\R)$ for which all orbit closures are finite volume almost homogeneous spaces, namely {\it iff\/} the smallest observable subgroup of $\SL_n(\R)$ containing $F$ has no nontrivial characters defined over $\R$.

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