Proper action on a homogeneous space of reductive type. (English) Zbl 0662.22008

An action of \(L\) on a homogeneous space \(G/H\) is investigated where \(L, H\subset G\) are reductive groups. A criterion of the properness of this action is obtained in terms of the little Weyl group of \(G\). Especially \(\mathbb{R}\)-\(\operatorname{rank}G= \mathbb{R}\)-\(\operatorname{rank} H\) iff Calabi-Markus phenomenon occurs, i.e. only finite subgroups can act properly discontinuously on \(G/H\). Then by using cohomological dimension theory of a discrete group, \(L\setminus G/H\) is proved compact iff \(d(G)=d(L)+d(H)\), where \(d(G)\) denotes the dimension of Riemannian symmetric space associated with \(G\), etc. These results apply to the existence problem of lattice in \(G/H\). Six series of classical pseudo-Riemannian homogeneous spaces are found to admit non-uniform lattice as well as uniform lattice, while some necessary condition for uniform lattice is obtained when \(\operatorname{rank} G = \operatorname{rank} H\).


22E40 Discrete subgroups of Lie groups
53C30 Differential geometry of homogeneous manifolds
43A85 Harmonic analysis on homogeneous spaces
53C35 Differential geometry of symmetric spaces
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