Timeline for Drinfeld Sokolov and the semiinfinite flag variety
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 12, 2021 at 21:10 | vote | accept | Pulcinella | ||
| Mar 11, 2021 at 19:46 | comment | added | David Ben-Zvi | And a caveat --- the word "twisted"here is slightly different than the standard one in D-module theory -- the Whittaker story is about twisting D-modules on G/N (or its loop version) by a G_a-torsor (character of N) rather than a G_m torsor (hence there are exponential maps around to relate it to more familiar twistings) | |
| Mar 11, 2021 at 19:44 | comment | added | David Ben-Zvi | In particular semi infinite flags DO seem to me an important part of the picture geoemtrically, except that we're studying a twisted form of D-modules on them rather than the untwisted one. Then the geometric operation @dhy is discussing is represented by this LG category of "twisted" D-modules on semiinfinite flags | |
| Mar 11, 2021 at 19:43 | comment | added | David Ben-Zvi | It might make sense to backtrack and make sure you're happy with the translation between algebraic and geometric interpretation of Whittaker for reductive groups say over C, before loop groups -- as a quantization of the Kostant slice eg, or as studying a twisted form of the cotangent bundle of G/N. The two have natural loop analogs, which are much more technical to make precise (cf Raskin's beautiful work) but morally speaking directly analogous. | |
| Mar 11, 2021 at 18:06 | answer | added | dhy | timeline score: 4 | |
| Mar 11, 2021 at 16:46 | comment | added | dhy | @Meow Aha, OK, that explains it. When I say Whittaker I mean the operation of taking $(LN,\chi)$-equivariant D-modules, defined e.g. in Section 1.13 of the paper of Raskin I linked earlier. | |
| Mar 11, 2021 at 15:43 | comment | added | Pulcinella | @dhy By Whittaker I mean the Whittaker functor as in Arakawa's ``Representation theory of W-algebras''. | |
| Mar 10, 2021 at 16:00 | comment | added | dhy | @Meow Well, my confusion is that you seem to be placing Whittaker on the algebraic side, while I think of it as a purely geometric construction. | |
| Mar 10, 2021 at 15:20 | comment | added | Pulcinella | @dhy Thanks for this, maybe baby Whittaker is what I'm after. I have no problem with Whittaker, but it is nice to understand things in as many ways as possible, and I've come to expect that often these things have both an algebraic and a geometric interpretation. | |
| Mar 8, 2021 at 3:43 | comment | added | dhy | @Meow Hmm, what is it about Whittaker itself that you don't like? In any case, here is one possible answer: For $X$ the affine flag variety (or the affine grassmannian,) Whittaker is the same as what is called "baby Whittaker," which is what is studied in e.g. Arkhipov-Bezrukavnikov (arxiv.org/abs/math/0201073). | |
| Mar 7, 2021 at 18:11 | comment | added | Pulcinella | @dhy Oh, that's great! Is there anywhere that I can see this written up? The ideal would be that, for a certain choice of $X$, the Whittaker functor on $D(X)$ corresponds to some other functor that is fairly geometric and is not a priori obviously the same as Whittaker (i.e. a "geometric construction which is not obviously the same as Whittaker"). | |
| Mar 7, 2021 at 13:26 | comment | added | dhy | @Meow Oh, if that's what you want then you're in luck: Whittaker is actually easier to describe on $D(X)$ than on $\hat{g}\operatorname{-mod}$. I more meant that there's no reason to take $X$ specifically the semi-infinite flag variety. | |
| Mar 6, 2021 at 10:40 | comment | added | Pulcinella | @dhy Thanks. I was aware of the Whittaker construction, so it's your last comment I'm most interested in. So if I understand it correctly you are saying that, although you can localise any $\widehat{\mathfrak{g}}$ module onto a bunch of different spaces $X$ (e.g. the semiinfinite flag variety), this doesn't interact with the Whittaker construction in the sense that there is no natural definition of the Whittaker functor acting on $Dmod(X)$ that isn't just ''consider a D module as a $\widehat{\mathfrak{g}}$ module and apply Whittaker". Is that right? | |
| Mar 4, 2021 at 23:08 | comment | added | dhy | P.S. The semi-infinite flag variety is an important part of this story, but I think for your question specifically it is a red herring. It appears when you think about $LN$ invariants without a twist. With a twist, the relevant geometry is instead that of Whittaker. This might seem like a small difference but it's really not. | |
| Mar 4, 2021 at 23:06 | comment | added | dhy | A more algebraic answer is given by A.2 of the cited paper. Raskin's heuristic explanation is that it is "cohomology along $\mathfrak{n}[[t]]$ and homology along $\mathfrak{n}((t))/\mathfrak{n}[[t]],$ and his definition makes this idea precise. My personal way of thinking about it is that it is "ordinary Lie algebra cohomology for $\mathfrak{n}((t))$, but shifted by $\operatorname{dim}\mathfrak{n}[[t]]$ degrees," which is complete nonsense (because $\mathfrak{n}[[t]]$ is infinite dimensional) but captures its behavior. | |
| Mar 4, 2021 at 22:59 | comment | added | dhy | A very satisfying answer is provided by arxiv.org/abs/1611.04937. In short, the Drinfeld-Sokolov functor comes from the 2-categorical functor of Whittaker coinvariants, applied to the category of Kac-Moody modules. However, this answer may be too categorical for your taste. | |
| Mar 4, 2021 at 9:49 | comment | added | Pulcinella | @Balazs Thanks for the response. Sadly I think in that book they don't actually give any Beilinson-Bernstein geometric explanation. The closest they get is that parenthetical remark about the classical limit being Hamiltonian reduction. | |
| Mar 4, 2021 at 9:17 | comment | added | Balazs | Have you looked at Frenkel--Ben-Zvi, Vertex algebras and algebraic curves, Chapter 16? I don't have the book at hand but I remember a fairly extensive discussion of Drinfeld Sokolov, likely with some geometric content given the style of the book. | |
| Mar 3, 2021 at 15:37 | history | asked | Pulcinella | CC BY-SA 4.0 |