The cubic Dirac operator on compact quotients of the oscillator group

TL;DR

This paper computes the spectrum of Kostant's cubic Dirac operator on certain Lorentzian manifolds formed by quotients of the four-dimensional oscillator group, providing explicit spectral and representation decompositions.

Contribution

It explicitly determines the spectrum and eigenspaces of the cubic Dirac operator on compact quotients of the oscillator group, including a detailed decomposition of the regular representation.

Findings

Spectrum of $D^{1/3}$ explicitly computed

Irreducible decomposition of the regular representation provided

Eigenspaces of $D^{1/3}$ explicitly characterized

Abstract

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 $D^{1/3}$.