Non-divergence of Translates of Certain Algebraic Measures

Let G and H ⊂ G be connected reductive real algebraic groups defined over Q, and admitting no nontrivial Q-characters. Let Γ ⊂ G(Q) be an arithmetic lattice in G, and π : G→ Γ\G be the natural quotient map. Let μH denote the H-invariant probability measure on the closed orbit π(H). Suppose that π(Z(H)) is compact, where Z(H) denotes the centralizer of H in G. We prove that the set {μH · g : g ∈ G} of translated measures is relatively compact in the space of all Borel probability measures on Γ\G, where μH · g(E) = μH(Eg) for all Borel sets E ⊂ Γ\G.