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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
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