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We show that, on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of finite characteristic with a given (generalized) regular central character are the same as coherent sheaves on the formal neighborhood of the corresponding (generalized) Springer fiber.The first step is to observe that the derived functor of global sections provides an equivalence between the derived category of D-modules (with no divided powers) on the flag variety and the appropriate derived category of modules over the corresponding Lie algebra. Thus the "derived" version of the Beilinson-Bernstein localization theorem holds in sufficiently large positive characteristic. Next, one finds that for any smooth variety this algebra of differential operators is an Azumaya algebra on the cotangent bundle. In the case of the flag variety it splits on Springer fibers, and this allows us to pass from D-modules to coherent sheaves. The argument also generalizes to twisted D-modules. As an application we prove Lusztig's conjecture on the number of irreducible modules with a fixed central character. We also give a formula for behavior of dimension of a module under translation functors and reprove the Kac-Weisfeiler conjecture.The sequel to this paper [BMR2] treats singular infinitesimal characters. To Boris Weisfeiler, missing since 1985 Contents

Abstract. In [BMR] we observed that, on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of finite characteristic with a given (generalized) regular central character can be identified with coherent sheaves on the formal neighborhood of the corresponding (generalized) Springer fiber. In the present paper we treat singular central characters.The basic step is the Beilinson-Bernstein localization of modules with a fixed (generalized) central character λ as sheaves on the partial flag variety corresponding to the singularity of λ. These sheaves are modules over a sheaf of algebras which is a version of twisted crystalline differential operators. We discuss translation functors and intertwining functors. The latter generate an action of the affine braid group on the derived category of modules with a regular (generalized) central character, which intertwines different localization functors. We also describe the standard duality on Lie algebra modules in terms of D-modules and coherent sheaves. §0. Introduction This is a sequel to [BMR]. In the first chapter we extend the localization construction for modular representations from [BMR] to arbitrary infinitesimal characters. This is used in the second chapter to study the translation functors. We use translation functors to construct intertwining functors, which generate an action of the affine braid group on the derived

To Joseph Bernstein with admiration and gratitude ROMAN BEZRUKAVNIKOV AND IVAN MIRKOVI Ć 4.2. Lifting to characteristic zero 4.3. Shifting the alcoves 5. Applications to Representation Theory 5.1. Generic independence of p 5.2. Equivariant versions and Slodowy slices 5.3. Gradings and bases in K-theory 5.4. Proofs for subsection 5.3 5.5. Koszul property 6. Grading that satisfies property (⋆)