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Journal of Lie Theory 31 (2021), No. 3, 833--869
Copyright Heldermann Verlag 2021

Tempered Homogeneous Spaces III

Yves Benoist
CNRS - Université Paris-Sud Orsay, Paris, France

Toshiyuki Kobayashi
Graduate School of Mathematical Sciences and Kavli IPMU (WPI), The University of Tokyo, Komaba, Japan

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 L2(G/H) is tempered. The proof gives the complete list of pairs (G,H) for which L2(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.

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