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Maybe this question is vague. I am not an expert on what I asked, if I made mistake, please point out.

BGG category was discoverd in Lie algebra setting. One has Verma module $M(\lambda)$, irreducible quotient $L(\lambda)$. Then one has the BGG-resolution, Weyl character formula, Kazhdan-Lusztig formula.

There are several generalizations:

  1. One can consider induction from parabolic subalgebra instead of Borel subalgebra to construct generalized Verma module. Then we still can consider analogue of BGG category O, BGG resolution, Kazhdan-Lusztig formula, so on so forth.

  2. One can construct BGG category O for quantum group.

  3. One can consider BGG category O for infinite dimensional flag variety, such as semi-infinite flag manifold.(see the work of Frenkel and Feign and geometric langland's school)

BGG category above are somehow related to Lie algebra. But there are still some analogue of BGG category which seems not related to Lie algebra.

  1. There is so called hyperbolic algebra or generalized Weyl algebra(for example, Weyl algebra, Heisenberg algebra, algebra of quantum differential operators). One can define BGG category O for this setting. We still have Verma module,irreducible quotient and so on. But I am not sure whether there is analogue of K-L formula.

  2. One can define BGG category for quiver variety. There is analogue of Kazhdan-Lusztig formula. See the notes of Nakajima.

My question is: BGG category appears everywhere and it seems that Kazhdan-Lusztig formula lives not only in Lie algebra setting but also many places else. This makes me guess that there should be some very general formula(general formalism)whose special case is K-L formula. I wonder whether anybody has investigated this phenomenon?

I know there are some work trying to do certain categorification of Verma module and associated irreducible quotient for quiver variety There is a paper by Zheng Hao on categorification of integral representation of quantum group. He considered $K_0$ of derived category of constructible sheaves on quiver variety and $K_0$ of its localization respect to thick subcategory. What he got is the quotient of K-group which is just irreducible integral representations. I think this might give some clue to look for general formalism behind Kazhdan-Lusztig formula.

There is another work by Peter Trapa, he considered the base change matrix between two bases of $K_0$ of BGG category O(one of them is Verma module, another one is associated irreducible quotient) He then discovered the same base change matrix between category of perverse sheaves and constructible sheaves. Then he tried to use K-L formula to give some geometric interpretation.

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Which formula are referring to? Kazhdan and Lusztig have produced a lot of formulas. –  Ben Webster Jun 5 '10 at 14:15
    
You also might want to be clearer on what you mean by "BGG category." There are precise notions like "highest weight category," but I have no idea if this is really what you mean. –  Ben Webster Jun 5 '10 at 14:50
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As Ben suggests, the question is overly broad. Some of the most classical theory is developed in my 2008 AMS book, but as you say there are now many "category $\mathcal{O}$" analogues. There have been efforts to study "highest weight categories" in some generality (Cline-Parshall-Scott), while Apoorva Khare has some relevant arXiv preprints including front.math.ucdavis.edu/0811.2080. But it's difficult if not impossible to fit all "Kazhdan-Lusztig conjectures" into one package. Different settings pose different technical challenges. –  Jim Humphreys Jun 5 '10 at 15:25
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Some other key adjectives: quasihereditary category, Koszul category. Jim's book and Ben's papers with Braden Licata and Proudfoot are good places to look. I also would recommend the (unique) paper by Mirollo and Vilonen explaining a geometric origin for BGG reciprocity in categories of perverse sheaves. –  David Ben-Zvi Jun 5 '10 at 23:39
    
The Mirollo-Vilonen paper in Ann. Sci. ENS 20 (1987) is freely available at www.numdam.org. On the other hand, to formulate a "Kazhdan-Lusztig" type of conjecture when a given category has unknown "simple" and known "standard" objects (related by a decomposition matrix), there is no known unifying idea covering all natural examples such as the Brauer-Nesbitt comparison of ordinary and p -modular characters of a finite group. – Jim 0 secs ago –  Jim Humphreys Jun 6 '10 at 14:16
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