Spectrum of semisimple locally symmetric spaces and admissibility of spherical representations

We consider compact locally symmetric spaces $\mathrm{\Gamma }\mathrm{\backslash }G\mathrm{/}H$ where $G\mathrm{/}H$ is a non-compact semisimple symmetric space and $\mathrm{\Gamma }$ is a discrete subgroup of $G$. We discuss some features of the joint spectrum of the (commutative) algebra $D(G\mathrm{/}H)$ of invariant differential operators acting, as unbounded operators, on the Hilbert space $L^2(\mathrm{\Gamma }\mathrm{\backslash }G\mathrm{/}H)$ of square integrable complex functions on $\mathrm{\Gamma }\mathrm{\backslash }G\mathrm{/}H$. In the case of the Lorentzian symmetric space $S{O}_0(2,2n)\mathrm{/}S{O}_0(1,2n)$, the representation theoretic spectrum is described explicitly. The strategy is to consider connected reductive Lie groups $L$ acting transitively and co-compactly on $G\mathrm{/}H$, a cocompact lattice $\mathrm{\Gamma }\subset L$, and study the spectrum of the algebra $D(L\mathrm{/}L\cap H)$ on $L^2(\mathrm{\Gamma }\mathrm{\backslash }L\mathrm{/}L\cap H)$. Though the group $G$ does not act on $L^2(\mathrm{\Gamma }\mathrm{\backslash }G\mathrm{/}H)$, we explain how (not necessarily unitary) $G$-representations enter into the spectral decomposition of $D(G\mathrm{/}H)$ on $L^2(\mathrm{\Gamma }\mathrm{\backslash }G\mathrm{/}H)$ and why one should expect a continuous contribution to the spectrum in some cases. As a byproduct, we obtain a result on the $L$-admissibility of $G$-representations. These notes contain the statements of the main results, the proofs and the details will appear elsewhere.