The cubic Dirac operator on compact quotients of the oscillator group

Abstract

We study Kostant’s cubic Dirac operator $D^{1/3}$ on locally symmetric Lorentzian manifolds of the form $\Gamma \backslash {\mathrm{Osc}}_1$, where ${\mathrm{Osc}}_1$ is the four-dimensional oscillator group and $\Gamma \subset {\mathrm{Osc}}_1$ is a cocompact lattice. These quotients are the only four-dimensional, compact Lorentzian $G$-homoge­neous spaces for a solvable but non-abelian Lie group $G$. We determine the spectrum of $D^{1/3}$. We also give an explicit decomposition of the regular representation of ${\mathrm{Osc}}_1$ on $L^2$-sections of the spinor bundle into irreducible subrepresentations and we determine the eigenspaces of $D^{1/3}$.