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

TL;DR

This paper extends the geometric representation theory of reductive Lie algebras in prime characteristic to singular central characters, using localization techniques and braid group actions on derived categories.

Contribution

It develops a localization framework for singular central characters and describes the action of the affine braid group via intertwining functors on derived categories.

Findings

Localization for singular central characters via sheaves on partial flag varieties.

Intertwining functors generate an affine braid group action.

Duality on Lie algebra modules expressed through D-modules and sheaves.

Abstract

In math.RT/0205144 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 as sheaves on the partial flag variety corresponding to the singularity of the character. These sheaves are modules over a sheaf of algebras which is a version of twisted crystalline differential operators, but is actually larger. 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.