most recent 30 from http://mathoverflow.net 2013-06-19T09:20:15Z http://mathoverflow.net/feeds/question/23001 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23001/is-there-a-good-account-of-d-affinity-and-localization-theorem-for-partial-flag-v Ben Webster 2010-04-29T16:09:53Z 2010-11-16T09:21:35Z

<p>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).</p> <p>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", <a href="http://www.google.com/url?sa=t&amp;source=web&amp;ct=res&amp;cd=1&amp;ved=0CAgQFjAA&amp;url=http%3A%2F%2Farxiv.org%2Fpdf%2F0906.1555&amp;ei=Wq3ZS9HID8L-8Abwy5Rs&amp;usg=AFQjCNFNykkDGVUXPdoJWlLvQy3xOXtuPQ&amp;sig2=fPyu1LoaAw-uMAmw3PtdhQ" rel="nofollow">this paper</a> 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 <a href="http://www.math.harvard.edu/~gaitsgde/grad_2009/Hotta.pdf" rel="nofollow">the book of Hotta, Takeuchi and Tanisaki</a>.</p> <blockquote> <p>Does anyone know a published source where this is proved?</p> </blockquote> <p>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.</p>

http://mathoverflow.net/questions/23001/is-there-a-good-account-of-d-affinity-and-localization-theorem-for-partial-flag-v/23024#23024 virk 2010-04-29T19:24:38Z 2010-04-29T19:24:38Z

<p>None of these quite fit the bill, but they might be a start:</p> <p>Thm. 1.9 in Differential Operators on Homogeneous Spaces III by Borho and Brylinski (don't miss the footnote!)</p> <p>Prop. 8.2.1 and Thm. 8.3.1 in Representation theory and D-modules on flag varieties by Kashiwara</p> <p>Prop. 3.5 and Thm. 3.8 in Differential Operators on Homogeneous Spaces I by Borho and Brylinski</p>

http://mathoverflow.net/questions/23001/is-there-a-good-account-of-d-affinity-and-localization-theorem-for-partial-flag-v/23075#23075 Victor Protsak 2010-04-30T05:21:02Z 2010-04-30T05:21:02Z

<p>That's a well-known sticky point, since some things do not quite work as advertised in singular case. Try Sec 3.7 of</p> <p>Holland, Martin P., Polo, Patrick. $K$-theory of twisted differential operators on flag varieties. Invent. Math. 123 (1996), no. 2, 377--414</p>

http://mathoverflow.net/questions/23001/is-there-a-good-account-of-d-affinity-and-localization-theorem-for-partial-flag-v/46146#46146 Simon Wadsley 2010-11-15T20:09:30Z 2010-11-16T09:21:35Z

<p>The answer is now yes, I think </p> <p><a href="http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.0896v2.pdf" rel="nofollow">http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.0896v2.pdf</a></p> <p>Edit: as requested: <a href="http://arxiv.org/abs/1011.0896" rel="nofollow">http://arxiv.org/abs/1011.0896</a></p>