Tempered homogeneous spaces IV

TL;DR

This paper characterizes when the regular representation of a complex semisimple Lie group G on the space of square-integrable functions over the homogeneous space G/H is tempered, based on the presence of regular elements in the orthogonal complement of h in g.

Contribution

It provides a precise criterion linking the temperedness of the regular representation to the existence of regular elements in the orthogonal complement of h in g.

Findings

The regular representation of G in L^2(G/H) is tempered if and only if the orthogonal of h in g contains regular elements.

The criterion connects geometric properties of the subgroup H with harmonic analysis on G/H.

The result extends understanding of tempered representations in the context of complex semisimple Lie groups.

Abstract

Let G be a complex semisimple Lie group and H a complex closed connected subgroup. Let g and h be their Lie algebras. We prove that the regular representation of G in $L^2(G/H)$ is tempered if and only if the orthogonal of h in g contains regular elements.