Spectrum of semisimple locally symmetric spaces and admissibility of spherical representations

Abstract

We consider compact locally symmetric spaces $\Gamma \backslash G/H$ where $G/H$ is a non-compact semisimple symmetric space and $\Gamma$ is a discrete subgroup of $G$. We discuss some features of the joint spectrum of the (commutative) algebra $D(G/H)$ of invariant differential operators acting, as unbounded operators, on the Hilbert space $L^2(\Gamma \backslash G/H)$ of square integrable complex functions on $\Gamma \backslash G/H$. In the case of the Lorentzian symmetric space $S{O}_0(2,2n)/S{O}_0(1,2n)$, the representation theoretic spectrum is described explicitly. The strategy is to consider connected reductive Lie groups $L$ acting transitively and co-compactly on $G/H$, a cocompact lattice $\Gamma \subset L$, and study the spectrum of the algebra $D(L/L\cap H)$ on $L^2(\Gamma \backslash L/L\cap H)$. Though the group $G$ does not act on $L^2(\Gamma \backslash G/H)$, we explain how (not necessarily unitary) $G$-representations enter into the spectral decomposition of $D(G/H)$ on $L^2(\Gamma \backslash G/H)$ 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.

The results were announced in June 2019 at the conference Representation Theory XVI held in Dubrovnik, Croatia. The paper will appear in a Contemporary Mathematics volume dedicated to the conference