Singular localization and intertwining functors for reductive Lie algebras in prime characteristic

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) {\em 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 $\lambda$ as sheaves on the partial flag variety corresponding to the singularity of $\lambda$. These sheaves are modules over a sheaf of algebras which is a version of twisted crystalline differential operators. We discuss {\em translation functors} and {\em 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 $\mathcal{D}$-modules and coherent sheaves