  We determine the spectrum of Kostant's cubic Dirac operator $D^{1/3}$ on
locally symmetric Lorentzian manifolds of the form $\Gamma\backslash {\rm
Osc}_1$, where ${\rm Osc}_1$ is the four-dimensional oscillator group and
$\Gamma\subset {\rm Osc}_1$ is a (cocompact) lattice. Moreover, we give an
explicit decomposition of the regular representation of ${\rm Osc}_1$ on
$L^2$-sections of the spinor bundle into irreducible subrepresentations and we
determine the eigenspaces of $D^{1/3}$.
