  In this paper, we prove that infinitesimal equivariant Chern-Connes
characters are well-defined. We decompose an equivariant index as a pairing of
infinitesimal equivariant Chern-Connes characters with the Chern character of
an idempotent matrix. We compute the limit of infinitesimal equivariant Chern-
Connes characters when the time goes to zero by using the Getzler symbol
calculus and then extend these theorems to the family case. We also prove that
infinitesimal equivariant eta cochains are well-defined and prove the
noncommutative infinitesimal equivariant index formula for manifolds with
boundary.
