Translates of some homogeneous measures in arithmetic quotients of linear algebraic groups.

Let G be a linear algebraic group over the rational numbers Q, H be an observable Q-subgroup and \Gamma be commensurable with the integral points of G. Given a sequence of elements g_n in G(R), we study the limiting behavior of g_nH(R)\Gamma in G/\Gamma. For a bounded open set O in H(R), we give a criterion on when g_nO\Gamma diverges topologically to infinity in G/\Gamma. And when the full orbit does not diverge, we classify all possible limiting measures.