Spectral symmetry of the Dirac operator in the presence of a group action

Fegan, H. D.; Steer, B.

doi:10.1090/S0002-9947-1993-1075381-X

Spectral symmetry of the Dirac operator in the presence of a group action
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by H. D. Fegan and B. Steer PDF

Trans. Amer. Math. Soc. 335 (1993), 631-647 Request permission

Abstract:

Let $G$ be a compact Lie group of rank two or greater which acts on a spin manifold $M$ of dimension $4k + 3$ through isometries with finite isotropy subgroups at each point. Define the Dirac operator, $P$, on $M$ relative to the split connection. Then we show that $P$ has spectral $G$-symmetry. This is first established in a number of special cases which are both of interest in their own right and necessary to establish the more general case. Finally we consider changing the connection and show that for the Levi-Civita connection the equivariant eta function evaluated at zero is constant on $G$.

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