Timeline for translation functors in parabolic category $\mathcal{O}$
Current License: CC BY-SA 3.0
13 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Feb 28, 2013 at 20:16 | comment | added | Vít Tuček | Done. The link to (hopefully) improved question is above. Thank you for your time, I greatly appreciate it. Also, let me use this opportunity to thank you for your wonderful book on category $\mathcal{O}$. It is a truly great source of information! | |
| Feb 28, 2013 at 20:11 | history | edited | Vít Tuček | CC BY-SA 3.0 |
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| Feb 28, 2013 at 19:43 | comment | added | Vít Tuček | I think a better approach is to revert my question to the original version and create a new one. | |
| Feb 28, 2013 at 18:52 | comment | added | Jim Humphreys | @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. | |
| Feb 28, 2013 at 14:41 | history | edited | Vít Tuček | CC BY-SA 3.0 |
added 3054 characters in body; edited tags; edited title
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| Feb 28, 2013 at 13:10 | history | edited | Vít Tuček |
edited tags
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| Feb 28, 2013 at 1:37 | comment | added | Jim Humphreys |
@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.
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| Feb 13, 2013 at 20:18 | answer | added | Jim Humphreys | timeline score: 1 | |
| Feb 13, 2013 at 13:52 | comment | added | Vít Tuček |
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.
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| Feb 13, 2013 at 0:54 | comment | added | Jim Humphreys | 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. | |
| Feb 13, 2013 at 0:50 | comment | added | Jim Humphreys | 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). | |
| Feb 13, 2013 at 0:04 | history | asked | Vít Tuček | CC BY-SA 3.0 |