Timeline for Fiber functor of category of D-module on affine Grassmannian.
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
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| Jun 12, 2010 at 9:51 | comment | added | JJH | Can you kindly answer another related question? Thanks. mathoverflow.net/questions/20265/… | |
| Jun 6, 2010 at 21:07 | comment | added | JJH | It seems !-crystal is the correct replacement of D-module on singular space. In this note, the writing seems very messy. I have no idea that, given a crystal on X, how to reconstruct D-module. What is the problem 1 in the hint? | |
| Jun 6, 2010 at 14:50 | comment | added | Ryan Reich | In Braverman's notes math.harvard.edu/~gaitsgde/grad_2009/Dmod_brav.pdf, section 7.3, this issue is discussed. You should especially read the exercises. As for the grassmannian, I am afraid that as far as I know, one must always define everything in this inductive way. One can, of course, make explicit the choice of the schemes Si; for example, take them to be unions of the closures of the orbits of G([[t]]), which is especially well suited for the geometric Satake problem. | |
| Jun 6, 2010 at 10:49 | comment | added | JJH | I'm actually quite confused by this kind of definition. On some singular variety, by Kashiwara's lemma, we do have some realization of D-module on it. But is it really the definition? We have so many realizations, which one you will take? Then on affine Grassmannian. D-module on affine Grassmannian, look more like some complicated construction, not really the definition. Is it possible to have some more intrinsic point of view? | |
| Jun 5, 2010 at 19:42 | comment | added | Ryan Reich | The affine grassmannian is a "strict ind-scheme": there is a sequence of closed subschemes $S_1 \subset S_2 \subset \dots \subset \operatorname{Gr}_G$ of which the grassmannian is the union. In fact, we can take them all to be projective schemes of finite type. For the purposes of geometric Satake, we declare a D-module (or perverse sheaf) to be supported on one of the finite-dimensional pieces. Kashiwara's theorem is used to ensure that this is independent of the choice of the $S_i$'s and that this makes sense when they are not necessarily smooth. | |
| Jun 5, 2010 at 19:18 | comment | added | JJH | May I ask how you think of D-module on affine Grassmannian? | |
| Jun 5, 2010 at 6:16 | comment | added | Qiaochu Yuan | Welcome to MO, Ryan! | |
| Jun 5, 2010 at 4:20 | history | edited | Ryan Reich | CC BY-SA 2.5 |
and again
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| Jun 4, 2010 at 23:53 | history | edited | Ryan Reich | CC BY-SA 2.5 |
Fix a notation
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| Jun 4, 2010 at 21:15 | history | edited | Ryan Reich | CC BY-SA 2.5 |
Try to fix the latex rendering; added 4 characters in body
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| Jun 4, 2010 at 19:52 | history | answered | Ryan Reich | CC BY-SA 2.5 |