Equivariant eta forms and equivariant differential $K$-theory

In this paper, for a compact Lie group action,we prove the anomaly formula and the functoriality of the equivariant Bismut-Cheeger eta forms with perturbation operators when the equivariant family index vanishes. In order to prove them, we extend the Melrose-Piazza spectral section and its main properties to the equivariant case and introduce the equivariant version of the Dai-Zhang higher spectral flow for arbitrary dimensional fibers.Using these results, we construct a new analytic model of the equivariant differential K-theory for compact manifolds when the group action has finite stabilizers only,which modifies the Bunck-Schick model of the differential K-theory. This model could also be regarded as an analytic model of the differential K-theory for compact orbifolds. Especially, we answer a question proposed by Bunke and Schick about the well-definedness of the push-forward map.