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

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 localization theorem for affine Lie algebra. I don't even know how to formulate it.

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

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

1. at Critical level
2. at noncritical level

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

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 localization theorem for affine Lie algebra. I don't even know how to formulate it.

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