On sets invariant under the action of the diagonal group

We consider the action of the (n - 1)-dimensional group of diagonal matrices in SL(n, ℝ) on SL(n, ℝ)/Γ, where Γ is a lattice and n ≥ 3. Far-reaching conjectures of Furstenberg, Katok-Spatzier and Margulis suggest that there are very few closed invariant sets for this action. We examine the closed invariant sets containing compact orbits. For example, for Γ = SL(n, ℤ) we describe all possible orbit-closures containing a compact orbit. In marked contrast to the case n = 2, such orbit-closures are necessarily homogeneous submanifolds in the sense of Ratner.