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.



Derived Push-Forward of Morphism of Perverse Sheaves and Translation Functors
Consider the collection of equivariant derived categories $D^b\_{P\times Q}(G)$ for $P,Q$ parabolics corresponding to collections $I,J$ of simple roots. Between these categories we have pushforward and pullback functors.


On the other hand there is a collection of categories given by bimodules:
$R^I-Mod-R^J$, where $R^I=\mathbb C[\mathfrak h^\*]^{W\_I}=H^\bullet\_P(pt)$. Between these categories we have restriction and extension of scalars functors.


Now taking equivariant hypercohomology 
$$H^\bullet\_{P\times Q}:D^b\_{P\times Q}(G)\rightarrow R^I-Mod-R^J$$
gives functors between these two collections of categories. It translates pushforward (pullback) into restriction (extension) of scalars. Moreover $H^\bullet\_{P\times Q}$ is fully faithful on semi-simple objects. I think there is a discussion of this picture in the introduction of Williamsons PhD thesis.


The upshot is, that $\pi\_\*$ is simply restriction of scalars! 
So using Soergel bimodules, at least in concrete examples, you can compute $Ext(\mathcal L,\mathcal L)$ and $\pi\_\*$ explicitly.


Finally let me mention, Schnürers beautiful formality [result](http://www.freidok.uni-freiburg.de/volltexte/4662/pdf/diss.pdf). It gives an equivalence 
$$D^b\_B(G/P)=per-Ext(\mathcal L,\mathcal L)$$
between the equivariant derived category and modules over the equivariant $Ext$ algebra of simple objects. It translates $\pi^\*$ and $\pi\_\*$ into tensoring and homing with the bimodule $Ext(\mathcal L, \pi^\* \mathcal L)$.
Hence it can in principle be used to extend the above description to the non semi-simple case.


