A Note on Equivariant Eta Invariants

Abstract

We prove the regulairty of equivariant eta functions near the origin. We also propose an equivariant version of the Cheeger-Chou index theorem on spaces with conelike singularities. 0. INTRODUCTION Let M be an odd-dimensional compact Riemannian spin manifold with a fixed spin structure, D the Dirac operator on M. The q function associated to D is defined by [1] (0.1) r(s,D)= (sign)A dimF(E.) where A runs over the nonzero eigenvalues of D and J7(E.) is the eigenspace of A. If T: M -, M is an isometry preserving the orientation and spin structure and d TD = D d T, where d T: 1(S(M)) -, F(S(M)) is the lift of d T: 1(TM) -JT(TM) to the spinors, then one can also define the equivariant q function [1, 5] by (0.2) 11T(S , D) = E(signA) Tr dIT(E,) In the first two sections we prove some basic properties of equivariant q functions and in ?3, we point out that a slight modification of Bismut-Cheeger [3] yields an equivariant index theorem for spaces with conelike singularities. Received by the editors April 28, 1989 and, in revised form, June 30, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 58G 10, 58G25.