Journal of Lie Theory Volume 6 (1996) 147–163 C ◯1996 Heldermann Verlag

Criterion for proper actions on homogeneous spaces of reductive groups

Toshiyuki Kobayashi

Communicated by M. Moskowitz

Abstract. Let M be a manifold, on which a real reductive Lie group G acts transitively. The action of a discrete subgroup Γ on M is not always properly discontinuous. In this paper, we give a criterion for properly discontinuous actions, which generalizes our previous work [6] for an analogous problem in the continuous setting. Furthermore, we introduce the discontinuous dual ⋔(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.

Toshiyuki Kobayashi Graduate School of Mathematical Sciences University of Tokyo Komaba, Meguro, Tokyo 153 Japan and Institut Mittag-Leffler The Royal Swedish Academy of Sciences Auravägen 17 S-182 62 Djursholm, Sweden toshi@ms.u-tokyo.ac.jp

Received December 28, 1995 and in final form June 29, 1996

ISSN 0949–5932 / $2.50 ^{_{C} ◯} Heldermann Verlag