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 Kähler potentials on a toric Fano manifold.