Timeline for Is there a category of representations of a simple Lie algebra on which its Weyl group naturally acts?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Dec 13, 2016 at 13:30 | answer | added | Bugs Bunny | timeline score: 1 | |
| Sep 17, 2013 at 17:54 | comment | added | Chuck Hague | @TobiasKildetoft - I don't know a lot about the functors involved, but my understanding is that twisting functors on $\mathcal O$ predate Arkhipov's functors, which he first defined in his 2001 paper "Algebraic construction of contragradient quasi-Verma modules in positive characteristic." | |
| Sep 13, 2013 at 7:00 | comment | added | Tobias Kildetoft | @ChuckHague Which functors do you refer to as Arkhipov's if not the twisting functors? | |
| Sep 12, 2013 at 22:59 | answer | added | Ben Webster♦ | timeline score: 15 | |
| Sep 12, 2013 at 14:43 | comment | added | Chuck Hague | There are a number of endofunctors on $\mathcal O$ that you may want to look at in addition to the twisting functors such as Arkhipov's, Enright's and Irving's functors (caveat: I am far from an expert on this!). Humphreys' book has some material on these functors; also see the papers "On Arkhipov's and Enright's functors" by Khomenko and Mazorchuk and "On functors associated to a simple root" by Mazorchuk and Stroppel. I think, though, that these functors only satisfy braid relations and not Weyl group relations, so this may not be quite what you're looking for. | |
| Sep 12, 2013 at 13:17 | comment | added | Tobias Kildetoft | BTW, I am usually in the representation theory chat room in case you want to discuss these topics. | |
| Sep 12, 2013 at 13:07 | comment | added | Tobias Kildetoft | I recall the section on twisting functors in Humphrey's book to be a bit on the short side. There is a paper by Andersen and Stroppel, "Twisting functors on $\mathcal{O}$" which I recall has a quite good introduction to the subject. | |
| Sep 12, 2013 at 12:55 | comment | added | John Baez | +Tobias Kildtoft - I'll need to learn about twisting functors and the relations they obey. My post included a link to a book by Humphreys. Is that a good place to learn this material, or should I go elsewhere? | |
| Sep 12, 2013 at 12:53 | comment | added | John Baez | I don't know what "do I want isomorphism classes of representations?" means. I want a category whose objects are some of the representations of $\mathfrak{g}$, on which the Weyl group has a natural (and nontrivial) action. | |
| Sep 12, 2013 at 10:56 | comment | added | Tobias Kildetoft | So first of all, one should probably consider just the principal block of $\mathcal{O}$, so the irreducibles are indexed by the Weyl group. The twisting functors come close, but they don't satisfy quite the correct relations when applied to the irreducibles. | |
| Sep 12, 2013 at 10:44 | comment | added | Marc Palm | Do you want isomorphism classes of representations? Are you aware that the intertwiner of parabolic induced representations of reductive groups over local fields are families of operators indexed by the Weyl group associated to the corresponding Levi-subgroup. | |
| Sep 12, 2013 at 10:35 | history | asked | John Baez | CC BY-SA 3.0 |