Spectra of Compact Quotients of the Oscillator Group

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