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 ${\rm Osc}_1$, which semi-direct product three-dimensional Heisenberg and real line. classify lattices Osc}_1$ up inner automorphisms Osc}_1$. For every lattice $L$ in we compute decomposition right regular representation on $L^2(L\backslash{\rm Osc}_1)$ into irreducible unitary representations. allows explicit computation spectrum wave operator locally-symmetric Lorentzian manifold $L\backslash {\rm