## Criterion for proper actions on homogeneous spaces of reductive groups.(English)Zbl 0863.22010

Summary: Let $$M$$ be a manifold, on which a real reductive Lie group $$G$$ acts transitively. The action of a discrete subgroup $$\Gamma$$ on $$M$$ is not always properly discontinuous. We give a criterion for properly discontinuous actions, which generalizes our previous work [the author, Math. Ann. 285, 249-263 (1989; Zbl 0672.22011)] for an analogous problem in the continuous setting. Furthermore, we introduce the discontinuous dual $$\pitchfork (H:G)$$ 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.

### MSC:

 22E40 Discrete subgroups of Lie groups 43A85 Harmonic analysis on homogeneous spaces 53C30 Differential geometry of homogeneous manifolds

Zbl 0672.22011
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