Criterion for Proper Actions on Homogeneous Spaces of Reductive Groups

Let M be a manifold, on which a real reductive Lie group G acts transitively.The action of a discrete subgroup on M is not always properly discontinuous.In this paper, we give a criterion for properly discontinuous actions, which generalizes our previous work [6] for an analogous problem in the continuous setting.Furthermore, we introduce the discontinuous dual (H:G) of a subset H of G , and prove a duality theorem that each subset H of G is uniquely determined by its discontinuous dual up to multiplication by compact subsets.