Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
add comment

1 Answer

up vote 1 down vote accepted

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.