I hope this question is not too vague.

Let $G$ be a complex reductive group, $B$ a Borel subgroup of $G$, and $P$ a parabolic containing $B$. Denote by $\pi:G/B\to G/P$ the canonical map. Consider the right derived functor $R\pi_*:D^b_B(G/B)\to D^b_B(G/P)$, $\mathcal{F}_1$ and $\mathcal{F}_2$ simple perverse sheaves, and $f\in \hom_{D^b_B(G/B)}(\mathcal{F}_1,\mathcal{F}_2[n])$ for some $n>0$.

** My Question: **
Are there any good functioning methods to compute $R\pi_* f$? Or have they already been computed?

Especially if $\mathcal{L}$ is the sum of simple perverse sheaves, then the algebra $Ext(\mathcal{L},\mathcal{L})$ is pretty well understood. Is it possible to use this knowlegde to compute $R\pi_*$ for the generators of $Ext(\mathcal{L},\mathcal{L})$?

** Variant: **
Of course one can replace $D^b_H(G/B)$ by $D^b_K(G/B)$, where $K$ is defined as in the book of Adams, Barbasch and Vogan, and then ask the same question.

** Motivation: **
By Koszul Duality the semi-simple objects of $D^b_B(G/P)$ (resp. $D^b_K(G/P)$) can be understood as the projective objects of the category $\mathcal{O}$ (resp. Harish-Chandra Modules, but here the definition of projective objects has to be modified), and the functors $R\pi_*$ and $\pi^*$ should somehow be the Koszul Dual of Translation functors. I would like to understand the latter statement.