Bismut-Cheeger eta form and higher spectral flow

In this paper, using the equivariant version of the Dai-Zhang higher spectral flow, we generalize the variation formula, embedding formula and the adiabatic limit formula for the Atiyah-Patodi-Singer eta invariants to the equivariant Bismut-Cheeger eta forms for a fiberwise compact Lie group action when the kernels of corresponding fiberwise Dirac operators form equivariant vector bundles.