Equivariant \(\eta\)-invariants on homogeneous spaces. (English) Zbl 0941.58016

The author computes explicitly equivariant \(\eta\)-invariants of equivariant Dirac operators \(D\) on quotients \(M=G/H\) of compact connected Lie groups \(G, H.\) A new operator \(\widetilde{D},\) the so-called reductive Dirac operator is defined, whose equivariant \(\eta\)-invariant \(\eta_G(\widetilde{D})\) is computed explicitly in terms of representation theoretic data of the groups \(G, H,\) and the embedding \(H \subset G.\) On the dense subset \(G_0\) of \(G\) which acts freely on \(M\), the equivariant \(\eta\)-invariants \(\eta_G(D)\) and \(\eta_G(\widetilde{D})\) differ only by a virtual character of \(G.\) These invariants coincide if the manifold is a symmetric space or if \(D\) is the untwisted Dirac operator.
Most of the results are also contained in [S. Goette, ‘Äquivariante \(\eta\)-Invarianten homogener Räume’, Diss., Shaker Verlag, Freiburg (1997; Zbl 0894.58068)].


58J28 Eta-invariants, Chern-Simons invariants
43A85 Harmonic analysis on homogeneous spaces


Zbl 0894.58068
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