Tempered homogeneous spaces III

TL;DR

This paper characterizes when the regular representation of a real semisimple Lie group G on the space of square-integrable functions over the homogeneous space G/H is tempered, focusing on the structure of stabilizers in complex cases.

Contribution

It provides a complete description of pairs (G,H) where the representation on L^2(G/H) is tempered, especially identifying the role of stabilizers in complex cases.

Findings

Temperedness characterized by virtually abelian stabilizers in complex cases.

Complete classification of (G,H) pairs with tempered representations.

Connection between group structure and representation properties.

Abstract

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. 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.