Spectral symmetry of the Dirac operator for compact and noncompact symmetric pairs (Q808506)

On a compact even-dimensional Riemannian spin manifold the spectrum of the Dirac operator is symmetric, i.e. the eigenspaces \({\mathcal V}_{\lambda}\) of the eigenvalue \(\lambda\) and the eigenspace \({\mathcal V}_{-\lambda}\) of the eigenvalue -\(\lambda\) are isomorphic. On the other hand the spectrum on an odd-dimensional manifold need not be symmetric. In this paper the following statement on the spectral symmetry of the Dirac operator is proved: Let (\({\mathcal G},{\mathcal K})\) be a symmetric pair of compact or non-compact type with rank at least two and \(\Gamma \subset {\mathcal G}\) be a co-compact discrete subgroup. Now consider a metric on \(\Gamma \setminus {\mathcal G}\) whose lift to \({\mathcal G}\) is \({\mathcal G}\)-left and \({\mathcal K}\)-right invariant. Then the eigenspace \({\mathcal V}_{\lambda}\) of the eigenvalue \(\lambda\) is \({\mathcal G}\)-isomorphic to the eigenspace \({\mathcal V}_{-\lambda}\) of the eigenvalue -\(\lambda\). In particular the equivariant \(\eta\)-function vanishes identically.