Spectra of elliptic invariant differential operators on locally symmetric spaces

Abstract

For a locally symmetric space $\Gamma \backslash G/K$ we study the spectrum of the invariant differential operators of $G/K$ on $L^2(\Gamma \backslash G/K)$. This gives a purely analytical proof of the recent theorem of arXiv:2402.02530 that the quasiregular representation $L^2(\Gamma \backslash G)$ is tempered if the limit cone ${\mathcal{L}}_{\Gamma }$ of $\Gamma$ is contained in the interior of the Weyl chamber except if $G$ has real rank one or is locally isomorphic to ${\mathfrak{sl}}_3(\mathbb{K})$, $\mathbb{K}=\mathbb{R},\mathbb{C},\mathbb{H}$, or ${\mathfrak{e}}_6^{-26}$.

26 pages, 6 figures