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Let us consider the category $\mathcal{O}$ for $sl_2$. For every central character $\theta$, we have the corrseponding block $\mathcal{O}(\theta)$. If we consider the block $\mathcal{O}(O)$ with trivial central character (the one in which highest weights can be $0$ or $-2$), it has a functor to the category of things of the form $(V_{-2} , V_0 , X: V_{-2} \to V_0 , Y: V_0 \to V_{-2})$ such that $XY = 0$. This functor is clearly exact and faithful.

My question (I would like an answer, or a reference...) are:

1) How to show that this functor is an equivalence (maybe first, how to define an adjoint).

2) What will be the corresponding quiver data for other blocks (probably $(V_n , V_{-n-2} , X^{n+1} , Y^{n+1}$) with some condition (?)).

3) What will be the corresponding assertions for a general semisimple group.

Thank you, Sasha

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Mazorchuk: Lectures on $\mathfrak{sl}_2(\mathbb{C})$-modules, section 5.3 is a reference. – Julian Kuelshammer Aug 29 '12 at 12:47
For other types of semi-simple Lie algebras, see e.g. Stroppel: Category $\mathcal{O}$: Quivers and Endomorphism rings of projectives. – Julian Kuelshammer Aug 29 '12 at 12:55
this chapter should be available with google books. – Julian Kuelshammer Aug 29 '12 at 12:56
It's worth looking at the papers authored or co-authored by Maxim Vybornov, most of them posted on the arXiv under AG, RT. (Maxim left academic work, but his collaborator Ivan Mirkovic is a good source. Catharina Stroppel's work is more current, also posted on arXiv.) – Jim Humphreys Aug 29 '12 at 13:45
Have you looked at ? – Bruce Westbury Aug 29 '12 at 15:07

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