Translates of homogeneous measures associated with observable subgroups on some homogeneous spaces

Abstract

In the present article we study the following problem. Let G be a linear algebraic group over Q, $\Gamma$ be an arithmetic lattice and H be an observable Q-subgroup. There is a H-invariant measure ${\mu }_H$ supported on the closed submanifold $H\Gamma /\Gamma$. Given a sequence $g_n$ in G we study the limiting behavior of $(g_n{)}_{\ast }{\mu }_H$. In the non-divergent case we give a rather complete classification. We further supplement this by giving criterion of non-divergence and prove non-divergence for arbitrary sequence $g_n$ for certain H. This work can be viewed as a natural extension of the work of Eskin--Mozes--Shah and Shapira--Zheng.

Reorganize the introduction; add one section discussing some applications; correct some typos/errors; main body of the paper stays the same