Category $\mathcal O$: Quivers and endomorphism rings of projectives

We describe an algorithm for computing quivers of category $\mathcal O$ of a finite dimensional semisimple Lie algebra. The main tool for this is Soergel's description of the endomorphism ring of the antidominant indecomposable projective module of a regular block as an algebra of coinvariants. We give explicit calculations for root systems of rank 1 and 2 for regular and singular blocks and also quivers for regular blocks for type $A_3$.

The main result in this paper is a necessary and sufficient condition for an endomorphism ring of an indecomposable projective object of $\mathcal O$ to be commutative. We give also an explicit formula for the socle of a projective object with a short proof using Soergel's functor $\mathbb V$ and finish with a generalization of this functor to Harish-Chandra bimodules and parabolic versions of category $\mathcal O$.