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 $\mathcal{O}_\mu$?


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

  • $\begingroup$ To deal with parabolic subcategories, you need to be more careful about your notation. This involves also weights and the notion of "facets" in the geometry related to Weyl chambers. As stated your question doesn't make sense. Origins go back to Jantzen's Habilitationsschrift: Lect. Notes. in Math. 750 (in German), but this along with later work is treated or referenced in my 2008 AMS book on the BGG category. In Chapter 9 is a detailed survey of parabolic subcategories. It would take extra work to adapt translation functors (Chapter 7) and get something new (or not). $\endgroup$ – Jim Humphreys Feb 13 '13 at 0:50
  • $\begingroup$ P.S. What Jantzen did originally uses ordinary 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
  • $\begingroup$ Thank you for your comments. What if I view $\mathcal{O}^\mathfrak{p}_\lambda$ 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
  • $\begingroup$ @robot: I can't follow what you mean by "restriction of the translation functor", which doesn't in general take the subcategory to itself. And your subscript on $L$ 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 1:37
  • $\begingroup$ @robot: Your original version and our comments plus my "answer" (an extended comment) make the new version hard to sort out. I don't know what MO protocol allows, but it might be simpler if we both delete the comments and my answer, so you can draft a better version of the question? By the way, I did include the 2006 arXiv post of the Boe-Hunziker paper (later published with additions) in the list of references in my book; but I gave up at that point trying to organize the fragmentary literature. $\endgroup$ – Jim Humphreys Feb 28 '13 at 18:52

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.


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