Spectra of compact quotients of the oscillator group (Q2035738)

If \(G\) is a Lie group and \(L\) is a cocompact discrete subgroup in \(G\), then we can consider the right regular representation of \(G\) on \(L^2(L\smallsetminus G)\), which is unitary. This representation can be decomposed into irreducible unitary representations. This paper aims to give a contribution to harmonic analysis of compact solvmanifolds. More precisely, one focus on the four-dimensional oscillator group \(\mathrm{Osc}_1\), which is a solvable Lie group, semi-direct product of the three-dimensional Heisenberg group and the real line. We classify the lattices of \(\mathrm{Osc}_1\) up to its inner automorphisms. For every lattice \(L\) in \(\mathrm{Osc}_1\), we compute the decomposition of the right regular representation of \(\mathrm{Osc}_1\) on \(L^2(L\smallsetminus\mathrm{Osc}_1)\) into irreducible unitary representations. AS a consequence, this decomposition allows the explicit computation of the spectrum of the wave operator on the compact locally-symmetric Lorentzian manifold \(L\smallsetminus\mathrm{Osc}_1\).