I hope the question can be reformulated more precisely, with all notation specified, since there may well be more to say about the parabolic subcategories of the BGG category $\mathcal{O}$. (Also, a tag lie-algebras would be appropriate.) But the notational conventions are rather complicated (and differ in Jantzen's original treatment from some choices I made). Translation functors by themselves are fairly innocuous at first, though of course simpler to work with once you assume all weights are *integral*. The deeper questions arise when you ask how translation functors behave on various classes of modules including Verma modules and then their simple quotients. It already takes some work to parametrize those modules by highest weights and keep the notation under control. (The question and comment involve confusing notation.)

For a fixed parabolic subalgebra (other than a Borel subalgebra or the full Lie algebra), everything is more complicated to set up and discuss. The starting point is a fixed subset of the simple roots. Then the relevant weights $\lambda$ of simple modules living in the subcategory are those which take nonnegative values at the given simple (co)roots. In order to restrict translation functors to the parabolic subcategory, you have to be sure that the functor preserves that category. Although the usual Verma modules are taken to others by translation functors, this isn't always the case in the parabolic setting. Moreover, the "blocks" of the parabolic category (in a strict sense) are not yet known in general, etc.

Originally the machinery was motivated by Lie group representation theory or related problems involving primitive ideals in the universal enveloping algebra of the Lie algebra. In my Chapter 9 survey of the parabolic case, I did mention in Remark (1) at the end of section 9.4 the discussion Jantzen included in section 2.25 of his 1979 *Lect. Notes in Math.* There the notation is even more intricate than what I use. Anyway, such fragmentary results may be all that is written down in the literature. Many loose ends remain. But it will require a lot of precise technique to extract more results from the limited algebraic machinery of translation functors. It might be instructive to work out some examples in rank 2 for "small" non-dominant weights.

central characters, but the broader work by Bernstein-Gelfand on projective functors relies on generalized characters of the center of the universal enveloping algebra. Notation is a bit tricky in some of the module categories encountered. $\endgroup$ – Jim Humphreys Feb 13 '13 at 0:54`$\mathcal{O}^\mathfrak{p}_\lambda$`

as a subcategory of`$\mathcal{O}_\lambda$`

and consider the restriction of the translation functor? My intended application is a cohomology formula for`$L_\mathfrak{p}(\lambda+\mu)$`

for $\mathfrak{g}$-integral $\lambda$ which is not`$\mathfrak{g}$`

-dominant. $\endgroup$ – Vít Tuček Feb 13 '13 at 13:52`$L$`

also doesn't make sense to me. There are no new simple modules in the subcategory. I think you need a more precise formulation in order to make contact with any of the literature. $\endgroup$ – Jim Humphreys Feb 28 '13 at 1:372more comments