# Is there a good account of D-affinity and localization theorem for partial flag varieties?

Recall that a topological space is called $A$-affine for a sheaf of algebras $A$ if taking global sections of coherent sheaves of $A$-modules is an equivalence of categories to finitely generated $\Gamma(A)$-modules. (for example, an affine scheme is one which is affine for the structure sheaf).

It seems to be a "well-known fact" that the variety $G/P$ for any simple complex algebraic group $G$ and parabolic $P$ is $D$-affine where $D$ is the sheaf of differential operators (and that more generally, one can quite explicitly describe the set of TDO's which are affine). I've found this stated in several books and papers (Beilinson and Bernstein's original paper, "Algebra V: homological algebra", this paper of Alexander Samokhin) but have yet to find an actual proof. One place one might guess it would be that it seems to not be is the book of Hotta, Takeuchi and Tanisaki.

Does anyone know a published source where this is proved?

I'll emphasize, what I want is not a proof of the theorem in the answers here; that's easy once you understand Beilinson and Bernstein's original argument. What I'm looking for in a place in the literature where this result is clearly and precisely stated, with a proof or clear reference to a proof.

• It makes one ponder that one has to ask for a reference for an actual proof of such a central result! Apr 30, 2010 at 5:48
• Mariano- Of course, the G/B case is the most popular, and is proved in Beilinson and Bernstein's paper. I think they just didn't have room for the G/P case (the paper's only 4 pages long, and covers a lot of ground). Apr 30, 2010 at 14:17

## 3 Answers

The answer is now yes, I think

http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.0896v2.pdf

Edit: as requested: http://arxiv.org/abs/1011.0896

• It is usually better to post a link to the abstract: arxiv.org/abs/1011.0896 Nov 16, 2010 at 2:57
• This recent preprint is intriguing but seems to require a lengthy detour through prime characteristic. Ben's original question should have a more direct answer (?), though probably not in the concise published form wanted. Anyway, the singular/parabolic interplay is important and still somewhat mysterious-looking to me; I guess it originates in Soergel's early work and independent work by Beilinson-Ginzburg. Nov 16, 2010 at 13:37

None of these quite fit the bill, but they might be a start:

Thm. 1.9 in Differential Operators on Homogeneous Spaces III by Borho and Brylinski (don't miss the footnote!)

Prop. 8.2.1 and Thm. 8.3.1 in Representation theory and D-modules on flag varieties by Kashiwara

Prop. 3.5 and Thm. 3.8 in Differential Operators on Homogeneous Spaces I by Borho and Brylinski

That's a well-known sticky point, since some things do not quite work as advertised in singular case. Try Sec 3.7 of

Holland, Martin P., Polo, Patrick. $K$-theory of twisted differential operators on flag varieties. Invent. Math. 123 (1996), no. 2, 377--414

• "What I'm looking for in a place in the literature where this result is clearly and precisely stated, with a proof or clear reference to a proof." (OP) - it doesn't seem so! May 4, 2010 at 19:16