Abstract
Bezrukavnikov, Mirkovic and Rumynin have recently obtained a derived version of the Beilinson-Bernstein localization theorem for the the Lie algebra of a semisimple algebraic group in positive characteristic p using the sheaf Open image in new window of rings of crystalline differential operators. A central reduction of Open image in new window is the first term Open image in new window of the p-filtration of the ring of the standard differential operators. We observe that the direct image functors of Open image in new window-modules on smooth varieties do not behave well; Kashiwara’s equivalence for a closed immersion fails, for example. On the other hand, we find that the direct image as Open image in new window-modules of the structure sheaf of the Frobenius neighbourhood of a point in each Chevalley-Bruhat cell under its inclusion in the flag variety realizes upon taking global sections an infinitesimal Verma module.
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