On the vanishing of the eta invariant of Dirac operators on locally symmetric manifolds
type: Item
instance of: scholarly article
title: On the vanishing of the eta invariant of Dirac operators on locally symmetric manifolds
zbMATH Open document ID: 1086.58012
author: Maria G. Gargova Fung
published in: Rocky Mountain Journal of Mathematics
publication date: 2006-01-24T00:00:00Z
review text: Let \(K\) be a maximal compact subgroup of a semi-simple Lie group \(G\). Assume \(G\) has no factors which are locally isomorphic to \(\text{SL}(3,\mathbb{R})\) or \(O(p,q)\) for \(p\) and \(q\) odd. Let \(\Gamma\) be a cocompact discrete torsion free subgroup of \(G\) and let \(M=\Gamma\backslash G\slash K\) be the associated local symmetric space. Assume that \(\dim(M)\) is odd and let \(\eta(D)\) be the eta invariant of the generalized Dirac operator on \(M\). \textit{H. Moscovici} and \textit{R. J. Stanton} [Invent. Math. 95, No. 3, 629--666 (1989; Zbl 0672.58043)] proved that \(\eta(D)=0\). The author gives a different proof of this result which is not based on an evaluation of the heat trace \(\text{Tr}_{L^2}\{De^{-tD^2}\}\) but which is modeled after the proof of the vanishing theorems of cohomology for the locally symmetric space \(M\) and is representation theoretic in nature. The author notes that the same proof extends to the case of the twisted \(\eta\)-invariant.
Mathematics Subject Classification ID: 58J28
zbMATH DE Number: 5001000
zbMATH Keywords: Eta invariant
zbMATH Keywords: local symmetric space
zbMATH Keywords: heat kernel
zbMATH Keywords: Clifford algebra
reviewed by: Peter B. Gilkey
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181069679
OpenAlex ID: W2010037671
cites work: A geometric construction of the discrete series for semisimple Lie groups
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cites work: Q3885948
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cites work: Closed geodesics and the \(\eta\)-invariant
cites work: Eta invariants of Dirac operators on locally symmetric manifolds
cites work: Q4320898
DOI: 10.1216/RMJM/1181069679
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description: scientific article; zbMATH DE number 5001000