Classification of irreducible symmetric spaces which admit standard compact Clifford-Klein forms. (English) Zbl 1416.53050

Tojo, Koichi

Proc. Japan Acad., Ser. A 95, No. 2, 11-15 (2019).

If \(G\) is a Lie group, \(H\) a closed subgroup of \(G\), \(\Gamma\) a discrete subgroup of \(G\), and \(\Gamma\) acts on a homogeneous space \(G/H\) properly, discontinuously, and freely, then the double coset space \(\Gamma\backslash G/H\) has a natural manifold structure. The double coset space \(\Gamma\backslash G/H\) with this manifold structure is called a Clifford-Klein form of \(G/H\) and \(\Gamma\) is called a discontinuous group for \(G/H\). If \(G/H\) is a homogeneous space of reductive type and \(\Gamma\) is a discontinuous group for \(G/H\), then a Clifford-Klein form \(\Gamma\backslash G/H\) is called standard if there exists a reductive subgroup \(L\) containing \(\Gamma\) and acting on \(G/H\) properly.
In this paper, the author gives a classification of irreducible symmetric spaces which admit standard compact Clifford-Klein forms. It is shown that if \(G\) is a linear noncompact semisimple Lie group and \(G/H\) is an irreducible symmetric space, and \(G/H\) admits a standard compact Clifford-Klein form, then \(G/H\) is locally isomorphic to a Riemannian symmetric space \(G/K\), or a group manifold \(G'\times G'/\text{diag}G'\), or one of the twelve homogeneous spaces admitting proper and cocompact actions of reductive subgroups \(L\).

Reviewer: Andrew Bucki (Edmond)

MSC:

53C30	Differential geometry of homogeneous manifolds
22F30	Homogeneous spaces
57M60	Group actions on manifolds and cell complexes in low dimensions
57S30	Discontinuous groups of transformations
22E40	Discrete subgroups of Lie groups
22E46	Semisimple Lie groups and their representations
53C35	Differential geometry of symmetric spaces

Keywords:

Clifford-Klein form; tangential homogeneous space; symmetric space; properness criterion; discontinuous group; real representation