Differential K-theory and localization formula for $\eta$-invariants

In this paper we obtain a localization formula in differential K-theory for $$S^1$$-actions. We establish a localization formula for equivariant $$\eta $$-invariants by combining this result with our extension of Goette’s result on the comparison of two types of equivariant $$\eta $$-invariants. An important step in our approach is to construct a pre-$$\lambda $$-ring structure in differential K-theory.