Comparison of two equivariant $\eta$-forms

In this paper, we first define the equivariant infinitesimal $\eta$-form, then we compare it with the equivariant $\eta$-form, modulo exact forms, by a locally computable form. As a consequence, we obtain the singular behavior of the equivariant $\eta$-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 $\eta$-invariants and on the differential $K$-theory.