Characteristic classes for $P \rightarrow G \rightarrow G/P$

Question

Let $G$ be a complex semisimple Lie group and let $P$ be a parabolic subgroup. We know that the cohomology of the flag variety $G/P$ is generated by Schubert classes.

There is a principal $P$ bundle, $$ P \rightarrow G \rightarrow G/P$$

Is it possible to give a general description of the characteristic classes of this bundle? If not, is it possible for $G=SL_n$ and all complex flag manifolds?