  In this paper, we establish an infinitesimal equivariant index formula in the
noncommutative geometry framework using Greiner's approach to heat kernel
asymptotics. An infinitesimal equivariant index formula for odd dimensional
manifolds is also given. We define infinitesimal equivariant eta cochains,
prove their regularity and give an explicit formula for them. We also establish
an infinitesimal equivariant family index formula and introduce the
infinitesimal equivariant eta forms as well as compare them with the
equivariant eta forms.
