Localization of modules for a semisimple Lie algebra in prime characteristic

TL;DR

This paper establishes a derived category equivalence between modules of a semisimple Lie algebra in large prime characteristic and coherent sheaves on Springer fibers, extending localization theory to positive characteristic.

Contribution

It proves a derived version of the Beilinson-Bernstein localization theorem in large prime characteristic and relates D-modules to coherent sheaves via Azumaya algebra splittings.

Findings

Derived equivalence between Lie algebra modules and coherent sheaves on Springer fibers.

Extension of localization theorem to characteristic p > h.

Calculation of Grothendieck group rank for modules with fixed central character.

Abstract

We observe that on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of characteristic $p> h$ (where $h$ is the Coxeter number), with a given (generalized) central character are the same as the coherent sheaves on (generalized) Springer fibers. 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, the algebra of (``crystalline'') differential operators is an Azumaya algebra and its splittings on Springer fibers allow us to pass from D-modules to coherent sheaves. As an application we compute the rank of the Grothendieck group of the category of modules over the Lie algebra with a fixed central character.