On the rate of exponential decay of coefficients on homogeneous spaces

TL;DR

This paper introduces a new exponent related to homogeneous spaces of noncompact semisimple Lie groups, providing a criterion for temperedness of certain representations that extends previous results.

Contribution

It defines an exponent with multiple interpretations and applies it to extend temperedness criteria for $L^2(G/H)$ beyond existing cases.

Findings

Introduces a new exponent with representation and group theory interpretations.

Provides a generalized temperedness criterion for $L^2(G/H)$.

Extends previous criteria to broader classes of subgroups.

Abstract

For any homogeneous space of a noncompact semisimple Lie group $G$, we define an exponent with multiple interpretations from representation theory and group theory. As an application, we give a temperedness criterion for $L^2 (G/H)$ for any closed subgroup $H$ of $G$, which extends the existing ones of Benoist--Kobayashi for connected subgroups and Lutsko--Weich--Wolf for discrete subgroups.