COMPACT CLIFFORD-KLEIN FORMS OF

PSEUDO-RIEMANNIAN HOMOGENEOUS MANIFOLDS

Toshiyuki KOBAYASHI (Univ. Tokyo & Univ. Lyon I)

Let G be a Lie group and H its closed subgroup. We say a discrete subgroup Γ of G is a discontinuous group for G/H if the natural action of Γ on G/H is properly discontinuous. If the action of Γ is furthermore fixed point free, then the double coset space Γ\G/H carries a natural manifold structure, which we say a Clifford–Klein form of a homogeneous manifold G/H. An important feature in our setting is that H is non-compact and that not all discrete subgroup of G can act properly discontinuously on G/H. Fundamental problems (cf. [Kb-1,2]) are:

(i) Which homogeneous manifold G/H admits an infinite discontinuous group? (ii) Which homogeneous manifold G/H admits a compact Clifford–Klein form?

Our concern is mainly with reductive cases where G/H carries naturally a pseudo-Riemannian metric, and we shall present (partly with survey nature):

(1) A solution of the so called Calabi-Markus phenomenon.

(2) A necessary and sufficient condition for discrete subgroups to act properly discontinuously on homogeneous manifolds of reductive groups.

(3) Relation with unitary representation theory and branching laws.

(4) A sufficient condition for the existence of compact Clifford-Klein forms.

(5) Sufficient conditions for the non-existence of compact Clifford-Klein forms.

(6) Deformation of compact Clifford-Klein forms.

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e-mail: toshi@ms.u-tokyo.ac.jp

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