On the rate of exponential decay of coefficients on homogeneous spaces
Yves Benoist, Siwei Liang
TL;DR
This work introduces four exponents—theta_G/H, delta_G/H, p_G/H, and beta_G/H—that quantify coefficient decay, volume growth, integrability, and local volume decay for homogeneous spaces G/H with G semisimple. It proves the central equalities theta_G/H = delta_G/H = 1 - 1/p_G/H ≥ beta_G/H, and shows that L^2(G/H) is tempered precisely when delta_G/H ≤ 1/2, unifying and extending prior criteria for both connected and discrete subgroups. The authors develop a robust framework linking representation-theoretic decay to geometric volume growth via rho-functions, Cartan and Iwasawa projections, and spherical functions, and they provide a detailed procedure to obtain uniform decay estimates for induced representations through a chain of intermediate subgroups inspired by Benoist–Kobayashi. The results yield explicit connections to Quint’s growth indicator for discrete subgroups and establish sharp relations between growth, decay, and integrability that refine our understanding of temperedness on general G/H.
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.