Spectra of Compact Quotients of the Oscillator Group

TL;DR

This paper classifies lattices in the 4D oscillator group, decomposes the regular representation, and explicitly computes the spectrum of the wave operator on associated compact Lorentzian manifolds.

Contribution

It provides a complete classification of lattices in the oscillator group and explicit spectral analysis of the wave operator on resulting compact quotients.

Findings

Classification of lattices in the oscillator group

Decomposition of regular representation into irreducibles

Explicit spectrum of the wave operator

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 $L\backslash {\rm Osc}_1$.