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Here is my question: how to define global section functor from D-module on affine flag variety to representation of affine Lie algebra?

Let's me explain the difficulty: it seems there doesn't exist global definition of D-module on ind-scheme. For affine flag variety, it is a union of finite dimensional subvarieties, and usually we can't make them smooth. We should think of a D-module on a singular variety as a usual D-module on big smooth space which supports on this singular variety. On the other hand, the global sections of D-module depends on the embedding of singular variety to the other smooth One.

I really don't know how to think of global section functor of D-module on affine flag variety, so I don't know how to formulate the localization theorem.

Maybe I should look at Frenkel-Gaitsgory's paper, but I'm afraid it is a question before reading their papers.

Moreover, I would like to know what is the status of localization theorem for affine Lie algebra? 1. at Critical level 2. at noncritical level

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Critical level: arxiv.org/abs/0712.0788 –  S. Carnahan Apr 3 '10 at 22:01
I was told by one student of Bernstein that even for affine Lie algebra(critical level), this localization type theorem is still not completed. For non-critical level, Kashiwara's definition of flag variety of Kac-Moody algebra is not well accepted(in some sense) and I did not know any work in this setting. –  Shizhuo Zhang Apr 3 '10 at 23:36
You would do quite a service to include in your question at least a statement of the terms used. –  Theo Johnson-Freyd Apr 4 '10 at 2:03
Let me get this straight: are you expecting others to explain you how to think of something that you cannot even formulate? What is the precise question? –  Victor Protsak Jun 12 '10 at 8:53
I don't know how to formulate the locallization theorem, that's because I don't know how to define the global section functor, that's exactly my question!! In the second paragraph, I explained the difficulty why I can not define it. It seems you haven't understood my question. –  xiyu Jun 12 '10 at 9:38
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1 Answer

up vote 2 down vote accepted

The main problem seems to be that you think the global section functor for (twisted) D-modules on a singular variety depends on a choice of embedding into a smooth variety. This is not true - D-modules can be defined on singular spaces using the infinitesimal site, and you can define global sections without any choice of embedding. Beilinson and Drinfeld describe the characteristic zero theory in section 7.10 of their unfinished book on Hitchin's integrable system, available from this page

Also, here are notes on D-modules on ind-schemes, from Dennis Gaitsgory's seminar.

The derived global section functor used in localization is constructed in section 23.5 of the Frenkel-Gaitsgory paper. If you read the introduction of the paper, you will find a statement of their results, and you will find a claim that much less is known away from the critical level.

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I should mention that the infinitesimal site is due to Grothendieck. –  S. Carnahan Jun 12 '10 at 22:26
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