Relative de Rham cohomology of flag varieties

Question

Let $B \subset P \subset G$ be a parabolic, Borel, and reductive (split) group over the complex numbers. Consider the projection $\pi: G/B \to G/P$, I am interested in computing $R\pi_{*} \Omega^{\bullet}_{G/B}$ in the derived category of coherent sheaves on $G/P$, where $\Omega^{\bullet}_{G/B}$ is the de Rham complex. More precisely, I think one should have $R\pi_{*} \Omega^{\bullet}_{G/B}=\Omega^{\bullet}_{G/P} \otimes H^{\bullet}(P/B,\mathbb{C})$, that is, copies of $\Omega^{\bullet}_{G/P}$ weighed by the singular cohomology group of $P/B$. My intuition behind this is that it seems that $H^{\bullet}(G/B)=H^{\bullet}(G/P) \otimes H^{\bullet}(P/B)$, using the combinatorics of Schubert calculus (at least in some low dimensional examples).

I am also think that this question is equivalent to computing the derived algebraic Induction from $P$ to $B$ of the Koszul complex of Verma modules.

Answer 1

Apologies for not having looked hard enough, but what I ask essentially follows from repeated application of the projective bundle formula in de Rham cohomology, at least for $G=\operatorname{GL}_n$.