Spectral analysis on standard locally homogeneous spaces

Let $X=G\mathrm{/}H$ be a reductive homogeneous space with $H$ noncompact, endowed with a $G$-invariant pseudo-Riemannian structure. Let $L$ be a reductive subgroup of $G$ acting properly on $X$ and $\mathrm{\Gamma }$ a torsion-free discrete subgroup of $L$. Under the assumption that the complexification $X_{\mathbb{C}}$ is $L_{\mathbb{C}}$-spherical, we prove an explicit correspondence between spectral analysis on the standard locally homogeneous space $X_{\mathrm{\Gamma }}=\mathrm{\Gamma }\mathrm{\backslash }X$ and on $\mathrm{\Gamma }\mathrm{\backslash }L$ via branching laws for the restriction to $L$ of irreducible representations of $G$. In particular, we prove that the pseudo-Riemannian Laplacian on $X_{\mathrm{\Gamma }}$ is essentially self-adjoint, and that it admits an infinite point spectrum when $X_{\mathrm{\Gamma }}$ is compact or $\mathrm{\Gamma }\subset L$ is arithmetic. The proof builds on structural results for invariant differential operators on spherical homogeneous spaces with overgroups.