# About localization theorem for affine Lie algebra?

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

• 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. – JJH Jun 12 '10 at 9:38