Goette, Sebastian Equivariant \(\eta\)-invariants and \(\eta\)-forms. (English) Zbl 0974.58021 J. Reine Angew. Math. 526, 181-236 (2000). 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. Reviewer: M.Raußen (Aalborg) Cited in 1 ReviewCited in 9 Documents MSC: 58J28 Eta-invariants, Chern-Simons invariants 58J20 Index theory and related fixed-point theorems on manifolds 43A85 Harmonic analysis on homogeneous spaces Keywords:eta-invariants; eta-forms; index; equivariant; Dirac operator Citations:Zbl 0297.58008; Zbl 0314.58016; Zbl 0321.58015; Zbl 0402.58006; Zbl 0941.58016; Zbl 0325.58015 PDFBibTeX XMLCite \textit{S. Goette}, J. Reine Angew. Math. 526, 181--236 (2000; Zbl 0974.58021) Full Text: DOI Link References: [1] Atiyah M. F., Math. Proc. Camb. Philos. Soc. 77 pp 43– (1975) [2] Atiyah M. F., Math. Proc. Camb. Philos. Soc. 78 pp 405– (1975) [3] Atiyah M. F., Math. Proc. Camb. Philos. Soc. 79 pp 71– (1976) [4] Atiyah M. F., Ann. Math. 92 pp 119– (1970) [5] S. Bechtluft-Sachs, On the h-invariant of Dirac operators on manifolds with free circle action, Shaker, Aachen 1993. · Zbl 0834.34101 [6] M. Berline, E. Getzler, N. Vergne, Heat kernels and Dirac operators, Springer, Berlin-Heidelberg-New York 1992. [7] Bismut J.-M., J. Funct. Anal. 62 pp 435– (1985) [8] Invent. Math. 83 pp 91– (1986) [9] Bismut J.-M., J. Am. Math. Soc. 2 pp 33– (1989) [10] Bismut J.-M., J. Funct. Anal. 89 pp 313– (1990) [11] Bismut J.-M., J. Funct. Anal. 90 pp 306– (1990) [12] Bismut J.-M., Pitman Monogr. Surv. Pure Appl. Math. 52 pp 59– (1991) [13] J.M. Bismut, S. Goette, Torsions analytiques eAquivariantes holomorphes, C. R. Acad. Sci. Paris 329 (I) (1999), 203-210. · Zbl 0952.53034 [14] J.M. Bismut, S. Goette, Holomorphic equivariant analytic torsions, preprint 1999, Geom. Funct. Anal., to appear. · Zbl 0974.58033 [15] Dai X., J. Am. Math. Soc. 4 pp 265– (1991) [16] Ch. Deninger, W. Singhof, The e-invariant and the spectrum of the Laplacian for compact nilmanifolds covered by Heisenberg groups, Invent. math. 78 (1984), 101-112. · Zbl 0558.55010 [17] Donnelly H., Indiana Univ. Math. J. 27 pp 889– (1978) [18] S. Goette, AEquivariante h-Invarianten homogener RaEume, Dissertation, Shaker Verlag, Aachen1997. [19] S. Goette, Equivariant h-Invariants on Homogeneous Spaces, Math. Z. 232 (1999), 1-42. · Zbl 0941.58016 [20] Hanson A. J., Phys. Lett. 80 pp 58– (1978) [21] Proc. Am. Math. Soc. 11 pp 236– (1960) [22] Math. 458 pp 37– (1995) [23] Duke Math. J. 82 pp 1– (1996) [24] Proc. Am. Math. Soc. 108 pp 1121– (1990) [25] Zhang W., Ann. Inst. Fourier 44 pp 249– (1994) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.