Kobayashi, Toshiyuki Criterion for proper actions on homogeneous spaces of reductive groups. (English) Zbl 0863.22010 J. Lie Theory 6, No. 2, 147-163 (1996). 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. Cited in 4 ReviewsCited in 30 Documents MSC: 22E40 Discrete subgroups of Lie groups 43A85 Harmonic analysis on homogeneous spaces 53C30 Differential geometry of homogeneous manifolds Keywords:manifold; real reductive Lie group; properly discontinuous actions; discontinuous dual; duality theorem Citations:Zbl 0672.22011 PDFBibTeX XMLCite \textit{T. Kobayashi}, J. Lie Theory 6, No. 2, 147--163 (1996; Zbl 0863.22010) Full Text: EuDML