Timeline for A precise statement of the categorical version of geometric Langlands conjecture
Current License: CC BY-SA 2.5
11 events
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| Oct 14, 2013 at 13:14 | comment | added | Dr. Evil | @David - Thanks for the usual informative posts. Your statement regarding the quantum version of local geometric Langlands puzzled me a bit. It seems that quantum Langlands is (again) mostly concerned with the unramified case. Recall that the FG proposal is that ones should attach to every ${}^L{G}$-local system a categorical representation of $G((t))$. Do you know a place where a quantum version of this proposal can be found? | |
| Feb 26, 2011 at 15:35 | comment | added | David Ben-Zvi | @Vinoth - there is, in the same documents that Tom talked about, though if the relevant ones are not available publicly on Dennis' page it's not my place to distribute them.. Dennis' paper is really about the local story, but in order to make technical sense of semiinfinite flags and Whittaker sheaves (which are the fundamental local category) one uses a standard ansatz in the story, using global curves (which are a bit of a red herring here).. | |
| Feb 26, 2011 at 15:32 | comment | added | David Ben-Zvi | @Kevin - thank you for the informative comments. Of course I agree that things are much vaguer in the ramified setting, except for functoriality, which can be formulated precisely with arbitrary ramification (it's a straightforward Tannakian exercise to formulate in the geometric setting). | |
| Feb 25, 2011 at 22:17 | comment | added | Puraṭci Vinnani | Thanks! One question: Is there a formulation of the quantum local geometric langlands conjecture, which you mention, anywhere? I think I once saw a talk Dennis gave on the subject, and Dennis's paper "Twisted Whittaker models & Factorizable Sheaves" seems to be on the "global" quantum geometric Langlands (unless I'm mistaken..). | |
| Feb 25, 2011 at 20:12 | comment | added | Kevin Buzzard | In particular I can not give a precise statement, which incorporates ramification, of any of the below things in the arithmetic setting: (1) functoriality (2) local Langlands (3) global Langlands, for a general $G$, even though I could talk all day about how things might work, and could make some more precise statements for $GL(n)$. I am just trying to make a "counter-point" to David's point: the OP asked for a precise conjecture, and I am suggesting that if you allow ramification then precise conjectures incorporating these phenomena are perhaps thinner on the ground! | |
| Feb 25, 2011 at 20:09 | comment | added | Kevin Buzzard | ...it is not clear to me that they would necessarily have a local conjecture, let alone a theorem, for general $G$. Maybe my comment that the geometric guys restrict to the unramified case when the going gets tough is unfair, but on the other hand it seems to me that David makes a precise global conjecture in the unramified case above, and in the ramified case it looks to my untrained eyes that his comments indicate that things are much vaguer---which in my mind is in perfect analogy with the arithmetic setting, where things are also vague. | |
| Feb 25, 2011 at 20:06 | comment | added | Kevin Buzzard | Thanks for the comments on ramification issues. On the arithmetic side there are two stories---one local and one global. In the local case the unramified story is understood completely. But the ramified story is not, except for some groups ($GL_n$, $GSp_4$...). I had initially thought that one might be able to formulate a local conjecture via global methods, but this also seems hard because globally you can have two $\pi$s that agree everywhere other than at one place, where they can be rather different. In particular even if the number theorists had functoriality in some great generality... | |
| Feb 25, 2011 at 17:01 | history | edited | David Ben-Zvi | CC BY-SA 2.5 |
Added discussion of ramification
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| Feb 25, 2011 at 4:09 | vote | accept | Puraṭci Vinnani | ||
| Feb 25, 2011 at 3:30 | history | edited | David Ben-Zvi | CC BY-SA 2.5 |
added 43 characters in body; added 444 characters in body
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| Feb 25, 2011 at 3:24 | history | answered | David Ben-Zvi | CC BY-SA 2.5 |