ag.algebraic geometry - What is the recent development of D-module and representation theory of Kac-Moody algebra?

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.