Abstract:  Let G be a semisimple linear algebraic group defined over rational numbers and Γ be an arithmetic lattice. One can associate a probability measure μ(H) on Γ\G for each subgroup H of G defined over Q with no non-trivial rational characters. As G acts on Γ\G from the right, we can push-forward this measure by elements from G. We call probability measures obtained this way homogeneous. It is a natural question to ask what are the possible weak-* limits of homogeneous measures. In the talk we will discuss some special cases that are close to unipotent dynamics.