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 $SO_0(2,2n)/SO_0(1,2n)$, 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.