I'm looking for a reference for a treatment of translation functors (as defined e.g. in this [question][1]) in parabolic versions of BGG category $\mathcal{O}$.

I am mainly interested in the following question.

Is it true that for integral weights $\lambda, \mu$ lying in the same facet one gets an equivalence of $\mathcal{O}_\lambda$ and $\mathcal{O}_\mu$?

Edit:

To avoid confusion I have created a new question under the title BGG-like resolutions and translations.

I'm looking for a reference for a treatment of translation functors (as defined e.g. in this [question][1]) in parabolic versions of BGG category $\mathcal{O}$.

I am mainly interested in the following question.

Is it true that for integral weights $\lambda, \mu$ lying in the same facet one gets an equivalence of $\mathcal{O}_\lambda$ and $\mathcal{O}_\mu$?

Edit:

To avoid confusion I have created a new question under the title BGG-like resolutions and translations.

I'm looking for a reference for a treatment of translation functors (as defined e.g. in this question) in parabolic versions of BGG category $\mathcal{O}$.

I am mainly interested in the following question.

Is it true that for integral weights $\lambda, \mu$ lying in the same facet one gets an equivalence of $\mathcal{O}_\lambda$ and $\mathcal{O}_\mu$?

Edit:

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.

Theorem 9.4 of Humphreys's Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$ says that the parabolic Verma modules (notation as in op.cit.) $M_I(\lambda)$ 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 $T_\lambda^\nu M_I(\lambda) \simeq M_I(\nu)$.

Cleaner/clearer argument is welcomed!

Let me try to change my question to the following.

Question: Let $I_1$, $I_2$ be subsets of simple roots and consider a BGG-like resolution of a simple module $L(\lambda)$ in $\mathcal{O}_{I_1}$. Under what conditions on $\lambda$ and $\mu$ can one "translate" this resolution to a resolution of $L(\mu)$ in $\mathcal{O}_{I_2}$? What are the known examples of such "translations"?

By a BGG-like resolution I mean a resolution in terms of direct sums of 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. Also note that Enright-Shelton equivalences are examples of such "translation".

I'm looking for a reference for a treatment of translation functors (as defined e.g. in this [question][1]) in parabolic versions of BGG category $\mathcal{O}$.

I am mainly interested in the following question.

Is it true that for integral weights $\lambda, \mu$ lying in the same facet one gets an equivalence of $\mathcal{O}_\lambda$ and $\mathcal{O}_\mu$?

Edit:

To avoid confusion I have created a new question under the title BGG-like resolutions and translations.

I'm looking for a reference for a treatment of translation functors (as defined e.g. in this question) in parabolic versions of BGG category $\mathcal{O}$.

I am mainly interested in the following question.

Is it true that for integral weights $\lambda, \mu$ lying in the same facet one gets an equivalence of $\mathcal{O}_\lambda$ and $\mathcal{O}_\mu$?

I'm looking for a reference for a treatment of translation functors (as defined e.g. in this question) in parabolic versions of BGG category $\mathcal{O}$.

I am mainly interested in the following question.

Is it true that for integral weights $\lambda, \mu$ lying in the same facet one gets an equivalence of $\mathcal{O}_\lambda$ and $\mathcal{O}_\mu$?

Edit:

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.

Theorem 9.4 of Humphreys's Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$ says that the parabolic Verma modules (notation as in op.cit.) $M_I(\lambda)$ 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 $T_\lambda^\nu M_I(\lambda) \simeq M_I(\nu)$.

Cleaner/clearer argument is welcomed!

Let me try to change my question to the following.

Question: Let $I_1$, $I_2$ be subsets of simple roots and consider a BGG-like resolution of a simple module $L(\lambda)$ in $\mathcal{O}_{I_1}$. Under what conditions on $\lambda$ and $\mu$ can one "translate" this resolution to a resolution of $L(\mu)$ in $\mathcal{O}_{I_2}$? What are the known examples of such "translations"?

By a BGG-like resolution I mean a resolution in terms of direct sums of 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. Also note that Enright-Shelton equivalences are examples of such "translation".