On the rate of exponential decay of coefficients on homogeneous spaces

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 $L^2(G\mathrm{/}H)$ 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.