BGG-like resolutions and translations

Question

This question originated from my confusion after I read the following paragraph (page 31, section 4.8) in Resolutions and Hilbert series of the unitary highest weight modules of the exceptional groups.

Suppose $\xi$ and $\nu$ lie in the positive chamber, have integral difference $\xi-\nu$ and equal stabilizers in $W$, i.e., $\{w\in W\, |\, w\xi = \xi\} = \{ w\in W\,|\, w\nu = \nu\}$. Applying standard techniques of Zuckerman translation we have an equivalence of categories $\Theta : \mathcal{O}_\xi \to \mathcal{O}_\nu$. Suppose $w\in W^c$ and assume both $w\xi$ and $w\nu$ are $\Delta_c^+$-dominant integral and regular. Then $\Theta N_{w\xi} \simeq N_{w\nu}$ and $\Theta L_{w\xi} \simeq L_{w\nu}$.

Notation here is that $\Delta_c^+$ is the set of positive roots of the Levi part $\mathfrak{l}$ and $W$ and $W^c$ are Weyl groups of $\mathfrak{g}$ and $\mathfrak{l}$ respectively. Symbol $N_\lambda$ denotes the parabolic Verma module induced from a $\Delta_c^+$-dominant integral weight $\lambda$ and $L_\lambda$ is it's simple quotient. Note that this article deals with $|1|$-graded situation only, so the nilradical of the parabolic is abelian.

These isomorphisms are then applied to a BGG-like resolution of unitarizable highest weight modules $L_\lambda$ and from the exactness of the translation functor $\Theta$ the conclusion is stated for invariance of some properties of these resolutions on "unitary strata".

I admit that the fact that translation functor maps parabolic Verma modules to parabolic Verma modules still seems a little bit unclear to me.

From now on I am going to use notation of Humphreys's Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$.

Theorem 9.4 of op.cit. says that the parabolic Verma modules $M_I(\lambda)$ fit into an exact sequence $$ \bigoplus_{\alpha\in I} M(s_\alpha\cdot \lambda) \to M(\lambda) \to M_I(\lambda) \to 0. $$ I guess that using the results on translation functors in section 7.8 coupled with the fact that homomorphisms of Verma modules are essentially unique one arrives at the conclusion $T_\lambda^\nu M_I(\lambda) \simeq M_I(\nu)$.

Q1: How to prove that in this simplest case the translation functor really maps parabolic Verma modules to parabolic Verma modules?

In general I am interested in BGG-like resolutions of simple modules in terms of direct sums of (parabolic) Verma modules. See e.g. Kostant modules in block of category $\mathcal{O}_S$ for examples of such resolutions outside of the classical BGG case.

Sometimes one can use exact functors to create whole families of such resolutions (see e.g. op. cit.). For example some infinitesimal blocks of the ordinary category $\mathcal{O}$ are equivalent to each other via translation functors which allows one to create whole families of resolutions of the same "shape".

Q2a: What are the known results on equivalences between infinitesimal blocks in parabolic category $\mathcal{O}_I$? What are the known results on translation functors in $\mathcal{O}_I$?

Q2b: What functors can be used for such a "translation"? What are known examples?

One example of 2b are Enright-Shelton equivalences constructed from derived translation functors that relate infinitesimal blocks of parabolic categories $\mathcal{O}$ for Lie algebras of different rank.

As was remarked by professor Humphreys in answer to my previous question, some results are contained in Jantzen's Moduln mit einem höchsten Gewicht.

Answer 1

One way to think about this picture is that the blocks (EDIT: infinitesimal blocks) of the full category $\mathcal{O}$ in this picture aren't just equivalent; they are equivalent in a way that preserves the labeling of simple modules by elements of the Weyl group. So, any module that has some nice characterization using these labels and categorical properties must be preserved. So, the projective cover $P_{w\xi}$ of a simple $L_{w\xi}$ must be sent to $P_{w\nu}$.

Similarly, the Verma module $M_{w\xi}$ must be sent to $M_{w\nu}$ since it is largest quotient of $P_{w\xi}$ where $L_{w'\xi}$ for $w'<w$ in Bruhat order is not a composition factor.

You can describe a parabolic Verma module in a similar way; being in parabolic category $\mathcal{O}$ is just a question about what your composition factors are; you can only have $L_{w\xi}$ for $w$ shortest in a left coset of $W_{I}$ and longest in a right coset for the stabilizer of $\xi$. We can characterize $N_{w\xi}$ as the maximal quotient of $M_{w\xi}$ which only has these composition factors, and thus it must be sent to $N_{w\nu}$.

Answer 2

Ben's somewhat informal answer to Q1 seems to address the main issue well (apart from evading the uncertain nature of "blocks"). If I read Jantzen's Satz 2.25 correctly, it includes this kind of information about translation functors and parabolic Verma modules in a more general but also more formal style.

Besides my cautionary comment to the proposer on Q2a, Q2b, I should re-emphasize the unknown status of "blocks" in the parabolic categories. Like others I used the term "block" in my comments, but in reality it gets rather complicated and affects the information one gets from translation functors. I summarized what could be said about blocks (defined in a strict sense) as of five years ago in section 9.15 of my book, but the known results still seem to be fragmentary.

The most satisfactory case involves regular integral weights, where the blocks are just what you'd expect and what is taken for granted in the question. So this case is the most transparent one. But talking about equivalences of blocks and the like requires more care in explaining exactly what a block is in the parabolic setting.