The cubic Dirac operator on compact quotients of the oscillator group

We study Kostant’s cubic Dirac operator D1/3 on locally symmetric Lorentzian manifolds of the form Γ₁ , where Osc₁ is the four-dimensional oscillator group and Γ₁ 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 D1/3 . We also give an explicit decomposition of the regular representation of Osc₁ on L² -sections of the spinor bundle into irreducible subrepresentations and we determine the eigenspaces of D1/3 .