Square integrability of regular representations on reductive homogeneous spaces

TL;DR

This paper investigates conditions under which the regular representation of a reductive homogeneous space is square integrable, providing criteria for unitary subrepresentations and implications for the existence of discrete series.

Contribution

It offers a sufficient condition for $L^2(G/H)$ to be a subrepresentation of $L^2(G)$ and proves the non-existence of discrete series for certain homogeneous spaces.

Findings

Provided a sufficient condition for $L^2(G/H)$ to embed into $L^2(G)$

Established criteria related to Benoist-Kobayashi functions for square integrability

Proved non-existence of discrete series in specific homogeneous spaces

Abstract

Let $G$ be a real reductive Lie group and $H$ a reductive subgroup of $G$. Benoist-Kobayashi studied when $L^2(G/H)$ is a tempered representation of $G$ and in particular they gave a necessary and sufficient condition for the temperedness in terms of certain functions on Lie algebras. In this paper, we consider when $L^2(G/H)$ is equivalent to a unitary subrepresentation of $L^2(G)$ and we will give a sufficient condition for this in terms of functions introduced by Benoist-Kobayashi. As a corollary, we prove the non-existence of discrete series for homogeneous spaces $G/H$ satisfying certain conditions.