Spectra of Compact Quotients of the Oscillator Group

Abstract

This paper is a contribution to harmonic analysis of compact solvmanifolds. We consider the four-dimensional oscillator group Osc₁, which is a semi-direct product of the three-dimensional Heisenberg group and the real line. We classify the lattices of Osc₁ up to inner automorphisms of Osc₁. For every lattice 𝐿 in Osc₁, we compute the decomposition of the right regular representation of Osc₁ on 𝐿²(𝐿∖Osc₁) 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∖Osc₁.