Strong stability of the homogeneous Levi bundle

Let $G$ be a connected semisimple linear algebraic group defined over an algebraically closed field. Let $P\,\subset\,G$ be a parabolic subgroup without any simple factor, and let $L(P)$ denote the Levi quotient of $P$. In this continuation of \cite{Bi}, we prove that the principal $L(P)$--bundle $(G\times L(P))/P$ over the homogeneous space $G/P$ is stable with respect to any polarization on $G/P$. When the characteristic of the base field is positive, this principal $L(P)$--bundle is shown to be strongly stable with respect to any polarization on $G/P$