Category $\mathcal{O}$ of semisimple Lie algebra has been understood very well. One can decompose the category into different blocks by central characters, and evey block is Noetherian and Aritian, and with finite many irreducible objects.

I would like to know the corresponding structure of category $O$ of Kac moody algebra, especially for affine Lie algebra. For infinite dimensional Lie algebra, the central of universal envoloping algebra is big and complicated. For Affine Lie algebra, maybe one can decompose the category by different levels. Then let's fix a level $k$, and consider the corresponding category $O_k$. For convenience, we only consider representation with integral weight decomposition.

My question is

- For different noncritical $k$, can we decompose $\mathcal{O}_k$ into different blocks, and each of then contains only one integral irreducible representation.
- For critical $k$, it should be even more complicated, how to think of it?
- How about general Kac Moody algebra, is there any general theorem on this decomposition?

nonintegralweights require special arguments using Jantzen's integral Weyl subgroups. It's important to be precise about what you mean here by "block", since the term is sometimes used more loosely. – Jim Humphreys Apr 23 '10 at 19:25