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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!

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Can you clarify: are you wanting to work in the Hecke context, i.e. B-equivariant $\mathcal{D}$-modules (in which case, many people on MO can tell you a lot, but I'll leave it to experts to do that) or injectives in the full category of quasicoherent $\mathcal{D}$-modules? The latter seems likely to be hard to describe. For example, passing to injective envelopes doesn't seem to preserve support. So I strongly doubt one can expect anything like the nice description in commutative algebra of indecomposable injectives... –  Thomas Nevins Feb 25 '11 at 18:57
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up vote 4 down vote accepted

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.

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Isn't the affine case just a very special case of Quillen's theorem that if A is a non-negatively filtered algebra of finite global dimension and the associated graded of A is also of finite global dimension, then the K-theory of A is the same as that of the 0th filtration piece? (I suppose it also uses the theorem that if A is commutative and $Spec(A)$ is smooth then A has finite global dimension.) –  Peter Samuelson Feb 26 '11 at 2:52
    
Yes, I think that's right. Chapter V of Weibel's "K-book" (math.rutgers.edu/~weibel/Kbook/Kbook.V.pdf) has a treatment of this result: see Theorem 6.4. –  Konstantin Ardakov Feb 26 '11 at 11:18
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