Stationary measures and invariant subsets of homogeneous spaces (II)

Let G be a real Lie group, Λ a lattice of G, μ a compactly supported probability measure on G, and Γ the subgroup generated by the support of μ. We prove that, when the Zariski closure of the adjoint group Ad (Γ ) is semisimple with no compact factor, every μ-ergodic μ-stationary probability measure on G/Λ is homogeneous. We also prove similar results for p-adic Lie groups.