Modified Futaki invariant and equivariant Riemann-Roch formula

In this paper, we give a new version of the modified Futaki invariant for a test configuration associated to the soliton action on a Fano manifold. Our version will naturally come from toric test configurations defined by Donaldson for toric manifolds. As an application, we show that the modified $K$-energy is proper for toric invariant K\"ahler potentials on a toric Fano manifold.