The cubic Dirac operator on compact quotients of the oscillator group
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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.