K\"ahler-Ricci flow on homogeneous toric bundles

Assume that $X$ is a homogeneous toric bundle of the form $G^{\mathbb{C}}{\times }_{P,\tau }F$ and is Fano, where $G$ is a compact semisimple Lie group with complexification $G^{\mathbb{C}}$, $P$ a parabolic subgroup of $G^{\mathbb{C}}$, $\tau :P\to (T^m{)}^{\mathbb{C}}$ is a surjective homomorphism from $P$ to the algebraic torus $(T^m{)}^{\mathbb{C}}$, and $F$ is a compact toric manifold of complex dimension $m$. In this note we show that the normalized K"{a}hler-Ricci flow on $X$ with a $G\times T^m$-invariant initial K"{a}hler form in $c_1(X)$ converges, modulo the algebraic torus action, to a K"{a}hler-Ricci soliton. This extends a previous work of X. H. Zhu. As a consequence we recover a result of Podest`{a}-Spiro.