
Journal of Lie Theory 31 (2021), No. 3, 833869 Copyright Heldermann Verlag 2021 Tempered Homogeneous Spaces III Yves Benoist CNRS  Université ParisSud Orsay, Paris, France yves.benoist@math.upsud.fr Toshiyuki Kobayashi Graduate School of Mathematical Sciences and Kavli IPMU (WPI), The University of Tokyo, Komaba, Japan toshi@ms.utokyo.ac.jp Let G be a real semisimple algebraic Lie group and H a real reductive algebraic subgroup. We describe the pairs (G,H) for which the representation of G in L^{2}(G/H) is tempered. The proof gives the complete list of pairs (G,H) for which L^{2}(G/H) is not tempered. When G and H are complex Lie groups, the temperedness condition is characterized by the fact that the stabilizer in H of a generic point on G/H is virtually abelian. Keywords: Lie groups, homogeneous spaces, tempered representations, unitary representations, matrix coefficients, symmetric spaces. MSC: 22E46; 43A85, 22F30. [ Fulltextpdf (259 KB)] for subscribers only. 