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)].

MSC:

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

Zbl 0894.58068
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