Derived Beilinson-bernstein Localization in Positive Characteristic Notes by Gufang Zhao

This is the notes for the author’s presentation in the student seminar. We follow the paper [BMR1] to talk about a derived equivalence between the representations of a semisimple Lie algebra in positive characteristic and the category of twisted D-modules supported generalized Springer fibers. We work over a field k, and consider a semisimple Lie algebra g. Let U = U(g) be the enveloping algebra. If k has characteristic zero, one has the BeilinsonBernstein Localization Theorem, which gives an equivalence of abelian categories. When k has positive characteristic, the sheaf of differential operators still makes sense. (There are different versions and we are using the version called crystalline differential operators.) In fact, fixing a character λ of the Cartan subalgebra h, one can consider the sheaf of twisted differential operators Dλ B on the flag variety B. One can still consider the localization functor L and the global sections functor Γ. They are functors between module categories over U(g) with the fixed central character λ, denoted by ModU(g)λ, and the category of sheaves of coherent modules over Dλ B, denoted by Coh D λ B.