Kahler-Ricci flow on homogeneous toric bundles

Assume that X is a homogeneous toric bundle of the form G(C) x (P,tau) F and is Fano, where G is a compact semisimple Lie group with complexification G(C), P a parabolic subgroup of G(C), tau : P -> (T-m)(C) is a surjective homomorphism from P to the algebraic torus (T-m)(C), and F is a compact toric manifold of complex dimension m. In this note, we show that the normalized Kahler-Ricci flow on X with a G x T-m-invariant initial Kahler form in c(1)(X) converges, modulo the algebraic torus action, to a Kahler-Ricci soliton. This extends a previous work of Zhu. As a consequence, we recover a result of Podesta-Spiro.