Derived Push-Forward of Morphism of Perverse Sheaves and Translation Functors

Question

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.

Answer 1

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. 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.