Square integrability of regular representations on reductive homogeneous spaces
Kazushi Maeda, Yoshiki Oshima
TL;DR
This work addresses when $L^2(G/H)$ can be realized as a subrepresentation of $L^2(G)$ for a real reductive group $G$ and reductive subgroup $H$, strengthening the Benoist–Kobayashi temperedness framework. It introduces a strict inequality condition, $\rho_g(Y) < 2\rho_q(Y)$ for all nonzero $Y$ in the maximal split subspace $a$, to guarantee that $L^2(G/H)$ is square integrable and thus embeds into $L^2(G)$. The paper also derives corollaries on the nonexistence of discrete series for certain homogeneous spaces and provides several explicit examples to illustrate the criterion, linking geometric size relations between $H$ and $G$ to harmonic-analytic properties. Overall, the results illuminate how a relatively
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.