Timeline for Equivalence between $\mathcal{D}_\lambda$ modules and $\mathcal{D}_{0}$ modules
Current License: CC BY-SA 4.0
14 events
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| Nov 24, 2020 at 11:16 | history | edited | Pulcinella | CC BY-SA 4.0 |
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| Nov 23, 2020 at 20:29 | comment | added | Geordie Williamson | You are right that this is in Lusztig's book, and this is the "modern" reference AFAIK. One thing that is not as well-known as it should be is that in the case of category O (say $N$-equivariance) this follows from Soergel's 1990 J|AMS paper in a very beautiful way. This is not mentioned in the paper but is commented on in the MathSciNet review. | |
| Nov 23, 2020 at 20:26 | comment | added | David Ben-Zvi | In that case, I think you want something like the parabolic-singular Koszul duality of Beilinson-Ginzburg-Soergel, or rather a combination of that with the Koszul duality as formulated by Soergel ("Koszul duality and Langlands' philosophy", cf Bezrukavnikov-Yun) relating flag varieties for Langlands dual groups. You might also look for something closely related at arxiv.org/abs/1904.01176 | |
| Nov 23, 2020 at 20:24 | comment | added | David Ben-Zvi | I think perhaps you mean N-integrable or maybe better B-equivariant D-modules? eg if $\lambda=0$ I don't know of any such result relating ALL D-modules on Langlands dual flag varieties | |
| Nov 23, 2020 at 17:27 | comment | added | Sam Hopkins | I think your edit is almost right but you want that $w\lambda-\lambda$ pairs integrally with coroots $\alpha^\vee$ not roots. | |
| Nov 23, 2020 at 17:25 | history | edited | Pulcinella | CC BY-SA 4.0 |
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| Nov 23, 2020 at 16:53 | comment | added | Sam Hopkins | Right: often "weight" is used synonymously with "integral weight"; but here I think it should mean the more inclusive thing (corresponding to infinite-dimensional representations). | |
| Nov 23, 2020 at 16:39 | comment | added | LSpice | Oh, right, so weight in the sense of "weight for some representation", not in the sense of "pairs integrally with all roots" (which is what's meant by 'integral' here, if I'm understanding "fundamental weight" correctly). I'm used to using 'weight' for the latter (e.g., "the quotient of the weight lattice by the root lattice is the fundamental group of the adjoint form"). | |
| Nov 23, 2020 at 16:25 | comment | added | Sam Hopkins | An integral weight is usually one that belongs to the $\mathbb{Z}$-span of the fundamental weights. If $\lambda$ itself is non-integral, then there might not be any non-identity $w\in W$ for which $w\lambda-\lambda$ is integral (think of $\lambda$ with a very small but nonzero length). | |
| Nov 23, 2020 at 16:13 | comment | added | LSpice | I can never keep the terminology straight. Does 'integral' mean 'lies in the root lattice'? Don't all Weyl-group elements satisfy that condition? | |
| Nov 23, 2020 at 16:12 | history | edited | LSpice | CC BY-SA 4.0 |
Weyl algebra -> Weyl group
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| Nov 23, 2020 at 15:40 | history | edited | Sam Hopkins | CC BY-SA 4.0 |
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| Nov 23, 2020 at 11:36 | history | edited | Pulcinella | CC BY-SA 4.0 |
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| Nov 22, 2020 at 11:18 | history | asked | Pulcinella | CC BY-SA 4.0 |