On Kashiwara’s equivalence in positive characteristic

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 [inline-graphic not available: see fulltext] of rings of crystalline differential operators. A central reduction of [inline-graphic not available: see fulltext] is the first term [inline-graphic not available: see fulltext] of the p-filtration of the ring of the standard differential operators. We observe that the direct image functors of [inline-graphic not available: see fulltext]-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 [inline-graphic not available: see fulltext]-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.