Hidden Symmetries and Spectrum of the Laplacian on an Indefinite Riemannian Manifold

Inspired by Sunada’s problem, we find a six dimensional, noncompact Γ-periodic Riemannian manifold that admits countably many discrete spectra of the Laplacian. This manifold also carries a three dimensional complex structure with indefinite Kähler metric. We observe a hidden symmetry in the sense that the automorphism group of the indefinite Kähler metric is larger than the group of Riemannian isometries. This very symmetry breaks a path to the theory of discontinuous groups for non-Riemannian manifolds and the theory of discrete decomposable branching laws of unitary representations.

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