Spectral analysis on standard locally homogeneous spaces

TL;DR

This paper establishes a spectral analysis framework for standard locally homogeneous spaces with pseudo-Riemannian structures, linking it to subgroup actions and branching laws, and proves self-adjointness and spectral properties of the Laplacian.

Contribution

It provides an explicit correspondence between spectral analysis on these spaces and on subgroup quotients using branching laws, under spherical complexification assumptions.

Findings

Pseudo-Riemannian Laplacian is essentially self-adjoint.

Infinite point spectrum exists for compact or arithmetic cases.

Spectral analysis reduces to branching laws for irreducible representations.

Abstract

Let $X=G/H$ 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 $\Gamma$ 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 $X_{\Gamma}=\Gamma\backslash X$ and on $\Gamma\backslash L$ via branching laws for the restriction to $L$ of irreducible representations of $G$. In particular, we prove that the pseudo-Riemannian Laplacian on $X_{\Gamma}$ is essentially self-adjoint, and that it admits an infinite point spectrum when $X_{\Gamma}$ is compact or $\Gamma\subset L$ is arithmetic. The proof builds on structural results for invariant differential operators on spherical homogeneous spaces with overgroups.