Spectra of elliptic invariant differential operators on locally symmetric spaces

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