SIGMA 17 (2021), 051, 48 pages      arXiv:1912.00050      https://doi.org/10.3842/SIGMA.2021.051

Spectra of Compact Quotients of the Oscillator Group

Mathias Fischer and Ines Kath
Institut für Mathematik und Informatik der Universität Greifswald, Walther-Rathenau-Str. 47, D-17489 Greifswald, Germany

Received September 28, 2020, in final form April 24, 2021; Published online May 13, 2021

Abstract
This paper is a contribution to harmonic analysis of compact solvmanifolds. We consider the four-dimensional oscillator group ${\rm Osc}_1$, which is a semi-direct product of the three-dimensional Heisenberg group and the real line. We classify the lattices of ${\rm Osc}_1$ up to inner automorphisms of ${\rm Osc}_1$. For every lattice $L$ in ${\rm Osc}_1$, we compute the decomposition of the right regular representation of ${\rm Osc}_1$ on $L^2(L\backslash{\rm Osc}_1)$ into irreducible unitary representations. This decomposition allows the explicit computation of the spectrum of the wave operator on the compact locally-symmetric Lorentzian manifold $L\backslash {\rm Osc}_1$.

Key words: Lorentzian manifold; wave operator; lattice; solvable Lie group.

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