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 $G$).
