This paper is the second part of a series of papers on noncommutativegeometry and conformal geometry. In this paper, we compute explicitly theConnes-Chern character of an equivariant Dirac spectral triple. The formulathat we obtain for which was used in the first paper of the series. Thecomputation has two main steps. The first step is the justification that the CMcocycle represents the Connes-Chern character. The second step is thecomputation of the CM cocycle as a byproduct of a new proof of the localequivariant index theorem of Donnelly-Patodi, Gilkey and Kawasaki. The proofcombines the rescaling method of Getzler with an equivariant version of theGreiner-Hadamard approach to the heat kernel asymptotics. Finally, as a furtherapplication of this approach, we computate the short-time limit of the JLOcocycle of an equivariant Dirac spectral triple.
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