Timeline for Reference request: Beilinson-Bernstein for finite-dimensional reps and category O
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Sep 12, 2014 at 20:01 | comment | added | Jim Humphreys | P.S. Note especially Remark 12.2.8 in the Hotta-Tanisaki book on the transition from the weight 0 to an arbitrary dominant weight. | |
| Sep 12, 2014 at 18:53 | history | edited | Frano Aleksi | CC BY-SA 3.0 |
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| Sep 12, 2014 at 16:40 | comment | added | Jim Humphreys | Yes, as Geordie points out the initial arguments only yield information about the principal block of $\mathcal{O}$, so twisted D-modules are needed as well. You can find Gaitsgory's 2005/2010 notes and many links including unfinished Beilinson-Drinfeld book, at math.harvard.edu/~gaitsgde/grad_2009 (but probably without getting much new insight into finite dimensional representations) | |
| Sep 12, 2014 at 15:57 | comment | added | Geordie Williamson | Having associated variety the zero section is the same thing as being a vector bundle with flat connection. Now $\pi_1(G/B)$ is trivial, and so there are only trivial bundles whose global sections give direct sums of the trivial rep. To get all reps (and recover Borel-Weil) one instead should consider twisted $D$-modules. One possible ref for all of this is Gaitsgory's notes on geometric rep theory (on his web-page I think). | |
| Sep 12, 2014 at 14:58 | comment | added | Jim Humphreys | It's true that the finite dimensional representations are not explicitly covered in the geometric setting, since they are well-studied by classical methods and probably not better understood by B-B localization. So you need to read between the lines. Even the literature on associated varieties may not be of direct help here. | |
| Sep 12, 2014 at 13:10 | comment | added | Frano Aleksi | Thanks - I’m somewhat familiar with both books and didn’t have the impression these results were explicitly mentioned in either of them, but I’ll have another look. | |
| Sep 12, 2014 at 12:46 | comment | added | Jim Humphreys | The standard reference book is: R. Hotta, K. Takeuchi, T. Tanisaki, D-modules, perverse sheaves, and representation theory. Translated from the 1995 Japanese edition by Takeuchi. Progress in Mathematics, 236. Birkhäuser Boston, Inc., Boston, MA, 2008. My AMS graduate text published also in 2008 gives mainly the algebraic preliminaries for category $\mathcal{O}$, followed by a detailed outline of the Beilinson-Bernstein arguments (without the details). | |
| Sep 12, 2014 at 11:31 | review | First posts | |||
| Sep 12, 2014 at 11:37 | |||||
| Sep 12, 2014 at 11:29 | history | asked | Frano Aleksi | CC BY-SA 3.0 |