The cubic Dirac operator on compact quotients of the oscillator group
Let \(\mathrm{Osc}_1\) be the oscillator group, that is, the only \(4\)-dimensional solvable but non-abelian connected Lie group having compact \(4\)-dimensional Lorentzian quotients by a lattice. Actually, \(\mathrm{Osc}_1\) is a semi-direct product of the \(3\)-dimensional Heisenberg group by the real line. Given any cocompact lattice \(\Gamma\subset\mathrm{Osc}_1\) and any bi-invariant Lorentzian metric on \(\mathrm{Osc}_1\), the compact homogeneous manifold \(\Gamma\backslash\mathrm{Osc_1}\) is locally symmetric. In case the lattice is basic (Definition 4.1), then that lattice is isomorphic to the direct product of some discrete Heisenberg group with \(\mathbb{Z}\); moreover, the quotient \(\Gamma\backslash\mathrm{Osc_1}\) is spin and carries several spin structures described by homomorphisms \(\Gamma\to\mathbb{Z}_2\), see Section 4. On the associated spinor bundle, there is a \(1\)-parameter family of metric connections -- but which are not torsion-free, unless \(t=\frac{1}{2}\) -- and thus of Dirac operators \(D^t\) where \(t\in\mathbb{R}\), see Section 2. For \(t=\frac{1}{3}\), the operator \(D^t\) is the so-called cubic Dirac operator introduced by \textit{B. Kostant} [Duke Math. J. 100, No. 3, 447--501 (1999; Zbl 0952.17005)].\N\NThe authors prove the following, as is summarised at the end of Section 1. First, for any \(t\in\mathbb{R}\), the spectrum of \(D^t\) coincides with the complex line, see Theorem 6.9. Second, the residual spectrum of \(D^t\) is empty, see Theorem 6.9 as well. Third, the point spectrum of \(D^{\frac{1}{3}}\) is discrete and contains only real and purely imaginary numbers depending on the spin structure, see Section 6.2. Moreover, the eigenspaces of \(D^{\frac{1}{3}}\) can be written down explicitly, see Section 6.2. Fourth, the point spectrum of \(D^t\) can be computed in terms of the eigenvalues of \(D^{\frac{1}{3}}\), see Section 6.6.\N\NAs the authors mention, for arbitrary lattices, it is no longer true that the point spectrum of \(D^{\frac{1}{3}}\) is always discrete. In Section 6.5, examples of shifted (non-basic) lattices are given for which the point spectrum of \(D^{\frac{1}{3}}\) on the quotient has accumulation points.
scientific article; zbMATH DE number 7880774
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