Two versions of parabolic category O

Question

Let $ \mathfrak{g} $ be a semisimple Lie algebra with corresponding complex semisimple group $ G$. Let $ P \subset G $ be a parabolic subgroup. Let $ W^P $ be the set of shortest coset representatives for $ W_P$, in other words $ W^P = \{ w \in W : ws_j > w \text{ for all } j \in J \} $ where $ J \subset I $ is the set of simple roots for the Levi $L$ of $ P $.

I would like to ask about two versions of parabolic category $ \mathcal O $, or more precisely its principal block.

In the usual version, we ask for modules which lie in category $ \mathcal O $, but are locally finite for $ \mathfrak p $. In the principal block, this means that we consider the Serre subcategory generated by the simple objects $ L(w^{-1} \cdot 0)$ for $ w \in W^P $. (Note that if $ \lambda = w^{-1} \cdot 0 $, then $ \langle \alpha_j, \lambda \rangle = \langle \alpha_j, w^{-1} \rho - \rho \rangle = \langle w \alpha_j, \rho \rangle - 1 $ and since $ w s_j > w $, we see that $ w \alpha_j $ is a positive root and so $ \langle \alpha_j, \lambda \rangle \ge 0 $ and hence $ \lambda $ is $L$-dominant.)

On the other hand, the action of $ G $ on the partial flag variety $ G/P $ gives us a map $ U(\mathfrak g) \rightarrow D(G/P) $ and we can look at those objects in the principal block of category $ \mathcal O $ which are pulled back via this map.

Does this subcategory contain $ L(w \cdot 0) $ for $ w \in W^P$ ? (Note the change from above --- here I want $ L(w \cdot 0) $ instead of $ L(w^{-1} \cdot 0) $, so we get a different set of simples.)

It seems that these two versions of parabolic category $ \mathcal O $ are related by comparing $ D^{P}(G/B) $ and $ D^B(G/P)$, but I don't know how to make this precise.

(There is also maybe something going on here related to the two versions of category $ \mathcal O $ where we switch between generalized weight spaces / precise central character and precise weight spaces / generalized action of the centre, but I don't see how this can possibly give us this change in the highest weights of simples.)

This question is motivated by the theory of highest weight modules for truncated shifted Yangians. In type A, the map $ U(\mathfrak{sl}_n) \rightarrow D(G/P) $ can be realized as $ U (\mathfrak{sl}_n) \rightarrow Y \rightarrow Y^\lambda_0(R) \cong D(G/P)$ where $ Y $ is the Yangian and $Y^\lambda_0(R) $ is the truncated Yangian defined by a partition $ \lambda $ of $ n$ (which depends on $ P $) and a choice of parameters $ R $ (which is linked to the principal block). Then in our work, we classified all highest weights for modules for this $ Y^\lambda_0(R) $ and then we can compute these highest weights from the viewpoint of $ U(\mathfrak{sl}_n) $.

Answer 1

This is exactly the statement of Proposition 2 in Singular blocks of parabolic category O and finite W-algebras. The key observation is that inverting $w$ swaps the left and right cell orders on $W$.

EDIT: That was probably a little cavalier; if you are actually just focused on the question of "is $L(w\cdot 0)$ the sections of a D-module on $G/P$," you don't need to know the results in my paper to see it: it's the sections of the unique simple D-module which is the intermediate extension of the trivial local system on $BwP/P$.

The Milicic and Soergel functor referenced helps explain about the central character v.s. weight issue, though and clarifies what's going on with the geometry. This functor is roughly "pullback to G, apply inverse, and push forward to G/B" so this is sent to the intermediate extension of the trivial local system on $Pw^{-1}B/B$, which is of course, in parabolic category O. The inverse appears because being in parabolic category O has to with the left action of $P$, whereas being on G/P has to do with right, so of course we need an inverse to switch them.

The thing we have to be careful about is that a general Schubert smooth D-modules on $G/P$ will not be $B$-equivariant, so when we pushforward to G/B we get not a usual D-module, but rather a monodromic one (which algebraically, means that the center doesn't act by a scalar but potentially by something with a non-trivial nilpotent part).