Is there any work describing (indecomposable)projectives, injectives in category of D-modules on some flag variety?


I wonder whether someone has used quivers(say Auslander-Reiten sequences)to describe the homological properties for category of D-modules 


Thanks!



Descriptions of projectives (injectives) in category of D-modules
The following paper of Tim Hodges may be of interest: 


$K$-theory of $\mathcal{D}$-modules and primitive factors of enveloping algebras of semisimple Lie algebras. Bull. Sci. Math. (2) 113 (1989), no. 1, 85–88.


He proves that the Quillen $K$-groups of the abelian category of coherent $\mathcal{D}\_X$-modules on any smooth complex quasiprojective variety $X$ (such as a flag variety) are the same as the corresponding $K$-groups of $X$ (or equivalently the category of coherent $\mathcal{O}\_X$-modules.) The proof proceeds via a reduction to the case when $X$ is affine, and then by considering the associated graded of $\mathcal{D}(X)$.


It follows from this that if $X = G/B$ is a flag variety then $K\_0(\mathcal{D}\_X)$ is a free abelian group of rank $|W|$, where $W$ is the Weyl group of $G$. It should be possible to construct an explicit set of $|W|$ pairwise non-isomorphic indecomposable projective coherent $\mathcal{D}\_X$-modules using Schubert cells.


