Comparison of two equivariant η-forms

In this paper, we first define the equivariant infinitesimal η-form, then we compare it with the equivariant η-form, modulo exact forms, by a locally computable form. As a consequence, we obtain the singular behavior of the equivariant η-form, modulo exact forms, as a function on the acting Lie group. This result extends a result of Goette and it plays an important role in our recent work on the localization of η-invariants and on the differential K-theory.