Kavli Affiliate: Toshiyuki Kobayashi
| Summary:
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 $Γ$ 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 $X_Γ=Γbackslash X$ and on $Γbackslash L$ via branching laws for the restriction to $L$ of irreducible representations of $G$. In particular, we prove that the pseudo-Riemannian Laplacian on $X_Γ$ is essentially self-adjoint, and that it admits an infinite point spectrum when $X_Γ$ is compact or $Γsubset L$ is arithmetic. The proof builds on structural results for invariant differential operators on spherical homogeneous spaces with overgroups.
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