Non-divergence of translates of certain algebraic measures

Abstract.

Let G and $H\subset G$ be connected reductive real algebraic groups defined over $\Bbb Q$, and admitting no nontrivial $\Bbb Q$-characters. Let $\Gamma\subset G(\Bbb Q)$ be an arithmetic lattice in G, and $\pi:G\rightarrow\Gamma\backslash G$ be the natural quotient map. Let $\mu_H$ denote the H-invariant probability measure on the closed orbit $\pi(H)$. Suppose that $\pi(Z(H))$ is compact, where Z(H) denotes the centralizer of H in G. We prove that the set $\{\mu_H\cdot g :\in G\}$ of translated measures is relatively compact in the space of all Borel probability measures on $\Gamma\backslash G$, where $\mu_H\cdot g(E)=\mu_H(Eg^{-1})$ for all Borel sets $E\subset\Gamma\backslash G\$.