On the vanishing of the eta invariant of Dirac operators on locally symmetric manifolds
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.
scientific article; zbMATH DE number 5001000
scientific article