Non-divergence of reductive group action on homogeneous spaces

Let X=G/Γ be the quotient of a semisimple Lie group G by its non-cocompact arithmetic lattice. Let H be a reductive algebraic subgroup of G acting on X . We give several equivalent algebraic conditions on H for the existence of a fixed compact set in X intersecting every H -orbit. This generalizes previous results concerning certain special reductive group action on X in this setting. When G is of real rank one, Γ is a non-cocompact lattice of G , and $H<G$ is an algebraic group, we also obtain an algebraic condition on H which is equivalent to the return of every H -orbit to a single compact set in X . This complements our results in the case of an arithmetic lattice.