Spectral analysis on standard locally homogeneous spaces

2019/12/29 by Kassel, Fanny, Kobayashi, Toshiyuki
#11F72 #22E40 #22E46 #53C35 #58J50 #Differential Geometry (math.DG) #FOS: Mathematics #Representation Theory (math.RT) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1912.12601

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 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=Γ\backslash X and on Γ\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_Γ is essentially self-adjoint, and that it admits an infinite point spectrum when X_Γ is compact or Γ⊂ L is arithmetic. The proof builds on structural results for invariant differential operators on spherical homogeneous spaces with overgroups.

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