I just started to collect the papers of this field and know little things. So if I make stupid mistake, please correct me.

It seems that there are several approaches to localize Kac-Moody algebra(in particular, affine Lie algebra). I just took look at several papers by

**Kashiwara-Tanisaki:(1989)**

They constructed the flag variety of symmetrizable Kac-Moody algebra as ind-scheme.

**Edward Frenkel-B.Feign:**

They constructed the semi-infinite flag manifold and introduce the semi-infinite cohomology.

**Edward Frenkel-Dennis Gaitsgory:**

It seems that they dealt with two kinds of things:

1 Affine Grassmannian

2 Affine flag variety

**Olivier Mathieu**

It seems that he gave the general definition of flag variety of arbitrary Kac-Moody algebra as a stack.(I can read French but not very quickly, so there might be possibilities that I made a mistake to describe his work)

# My Question

Is there any other definition of flag variety of Kac-Moody algebra(at least for affine case)? What are the relationship between these definitions I mentioned above?

What is the relationship between the D-module theory on affine flag variety and D-module theory on affine grassmannian? (Frenkel-Gaitsgory) Why did they consider these two ways to localize affine Lie algebra?

It seems all of the construction above are not very easy to deal with(ind-scheme,group ind-scheme which are not locally affine). Is there any existent work to define it as a locally affine space(classical scheme,algebraic space or at least locally affine stack with smooth topology)?

From the work of Frenkel-Gaitsgory, they built the derived equivalence between the category of D-modules and full subcategory of modules over enveloping algebra of affine Lie algebra. They claimed in their paper that one can not obtain the equivalence in abelian level. Is there any intuitive explanation for this?

I am looking forward to getting some guy who can explain the work of Oliver to me.

Thank you in advance!