Comparison of two equivariant $\eta$-forms

In this paper, we define first 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 Goette's previous result to the most general case, and it plays an important role in our recent work on the localization of $\eta$-invariants.