What is the recent development of D-module and representation theory of Kac-Moody algebra? 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!
 A: I am not an expert in this but I would of course expect something like ind-scheme approach to be natural. Gerd Faltings used I think ind-schemes to treat Sugawara construction, algebraic loop groups and Verlinde's conjecture in 
Gerd Faltings, Algebraic loop groups and moduli spaces of bundles. 
J. Eur. Math. Soc. (JEMS) 5 (2003), no. 1, 41--68. 
You might also like to work with versions of Kac-Moody GROUPS in analytic approaches. 
You could also consult comprehensive and not that old Kumar's book (Kac-Moody groups, their flag varieties and representation theory, Birkhauser) which is written in geometric language. 
As far as Frenkel is concerned, not only his work with Feigin but even more I think his paper with Gaitsgory must be relevant (see arxiv:0712.0788). 
Semi-infinite cohomologies are important but still misterious thing. Some related homological algebra has been recently studied by Positelskii in great generality. Another important thing is relation between the geometry of representations of quantum groups at root of unity and of affine Lie algebras, like in the book of Varchenko and many papers later.
Edit: Frenkel himself I think does not claim (I talked to him at the time) to have intuitive explanation why only derived equivalence. But you should not expect for more: by the correspondence with quantum groups the situation should be like in affine case where one has problems with non-closedness of diagonal in noncommutative geometry what has repercussions on the theory of D-modules. How this reflects in the case of relevant ind-schemes for affine side I do not know but somehow it does.
