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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.

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