The cubic Dirac operator on compact quotients of the oscillator group

We study Kostant’s cubic Dirac operator D^1/3 on locally symmetric Lorentzian manifolds of the form ₁, where Osc₁ is the four-dimensional oscillator group and 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 Osc₁ on L^2 -sections of the spinor bundle into irreducible subrepresentations and we determine the eigenspaces of D^1/3.