The cubic Dirac operator on compact quotients of the oscillator group

Abstract

We study Kostant’s cubic Dirac operator D1/3 on locally symmetric Lorentzian manifolds of the form Γ\Osc1​, where Osc1​ is the four-dimensional oscillator group and Γ⊂Osc1​ 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.