Spectral symmetry of the Dirac operator for compact and noncompact symmetric pairs
type: Item
instance of: scholarly article
title: Spectral symmetry of the Dirac operator for compact and noncompact symmetric pairs
zbMATH Open document ID: 0732.58036
author: Howard D. Fegan
author: L. Whiteway
author: Brian Steer
published in: Pacific Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 4211127
zbMATH Keywords: Dirac operator
zbMATH Keywords: spectral symmetry
zbMATH Keywords: symmetric pair
zbMATH Keywords: equivariant \(\eta \) - function
reviewed by: Hans-Bert Rademacher
MaRDI profile type: Publication
full work available at URL: pjm.1990.145.211
OpenAlex ID: W2032317421
DOI: 10.2140/PJM.1990.145.211
description: scientific article; zbMATH DE number 4211127