  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 $(g_n)_*\mu_H$. 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.
