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This is a follow-up to this question; in particular, I'm wondering if anyone can expand upon the interesting answers given by Kevin McGerty and David Ben-Zvi there. (In particular, in this question I'm essentially quoting their answers).

Here's the setup: Let $k$ denote an algebraically closed field of positive characteristic and let $G$ be a semisimple algebraic group over $k$. Let $D$ denote the sheaf of ordinary differential operators on the flag variety $G/B$ of $G$; i.e., $D$ is the sheaf of divided-power differential operators. Also let $H$ denote the hyperalgebra of $G$.

Now, over $\mathbb C$ there is an equivalence of categories between $D$-modules and $H$-modules with a certain central character. My question is: Is there any sort of localization theorem like this in positive characteristic? Kashiwara and Lauritzen have shown that $G/B$ is not $D$-affine in general, so perhaps one should look for a derived equivalence. (Bezrukavnikov, Mirkovic, and Rumynin have answered a similar question, but instead of $D$ they take the sheaf of crystalline/PD differential operators, and instead of $H$ they take the enveloping algebra of the Lie algebra of $G$).

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As a general comment (which you should certainly feel free to ignore!) is that the sheaf of full differetial operators is not such a good one to work with. (Just as one example, if I remember correctly, the structure sheaf O_X on a smooth variety is not represented by a perfect complex of D-modules.) –  Emerton Feb 10 '10 at 18:23
This is the Lusztig Conjecture if I am not mistaken. –  B. Bischof Feb 10 '10 at 19:00
You may find what you're looking for in this article by C.Noot-Huyghe:… but the sheaf of differential operators involved is Berthelot sheaf of overconvergent p-adic differential operators. She proves G/B is D-affine for this sheaf of diff. operators. –  Laurent F. Feb 10 '10 at 22:00
Matthew - lack of perfection of the unit certainly makes life harder, but I think you have to roll with whatever you're given.. eg the same problem holds for BG_m or BG in characteristic zero. –  David Ben-Zvi Feb 12 '10 at 4:39
Hey BZ, Thanks for this remark, which is quite helpful. –  Emerton Feb 12 '10 at 6:17

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