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Equivariant \(\eta\)-invariants and \(\eta\)-forms. (English) Zbl 0974.58021

Let a Lie group \(G\) act isometrically on an odd dimensional Riemannian manifold \(M\) and let \(D\) be a \(G\)-equivariant operator on a \(G\)-equivariant Dirac bundle \(\mathcal{E}\) over \(M\).
The author constructs a \(\mathfrak{g}\)-equivariant \(\eta\)-invariant of \(D\) which is defined as a formal power series \(\eta _X(D)\) in \(X\in \mathfrak{g}\). It can be interpreted as a universal \(\eta\)-form for families of Dirac operators induced by \(D\) over fibrations with fibers isometric to \(M\) and with structure group \(G\). This \(\eta\)-form is then compared with the \(G\)-equivariant \(\eta\)-invariant of M. F. Atiyah, V. K. Patodi and I. M. Singer [Math. Proc. Camb. Philos. Soc. 77, 43-69 (1975; Zbl 0297.58008), ibid. 78, 405-432 (1975; Zbl 0314.58016), ibid. 79, 71-99 (1976; Zbl 0325.58015)] and H. Donnelly [Indiana Univ. Math. J. 27, 889-918 (1978; Zbl 0402.58006)]. For a small Killing field \(X\in \mathfrak{g}\) without zeros, the author obtains an explicit formula for the difference \(\eta_X(D)-\eta_{e^{-X}}(D)\) in terms of the equivariant \(\hat{A}\)-form of \(M\) and the equivariant relative Chern character form of \(\mathcal{E}\). For certain operators of Dirac type on the sphere, \(G\)-equivariant \(\eta\)-invariants have been calculated by Atiyah, Patodi and Singer; the author had recently obtained results for equivariant Dirac operators on homogeneous Dirac bundles over homogenoeus Riemannian spaces \(G/H\) [Math. Z. 232, 1-42 (1999; Zbl 0941.58016)]. In several of these cases, the author calculates the power series \(\eta _X(D)\) and especially its constant term \(\eta (D)\) by local means.

MSC:

58J28 Eta-invariants, Chern-Simons invariants
58J20 Index theory and related fixed-point theorems on manifolds
43A85 Harmonic analysis on homogeneous spaces
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References:

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