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 ${\mathrm{Osc}}_1$, 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 $L\backslash {\mathrm{Osc}}_1$.