Localization of modules for a semisimple Lie algebra in prime characteristic

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.

Bits and ends cleaned up, assumptions on p improved in some places. Appendix by Simon Riche and R.B. added

References in corpus (3)

• Frobenius splitting of cotangent bundles of flag varieties and geometry of nilpotent cones
• Duality and flat base change on formal schemes
• Geometric Representation Theory of Restricted Lie Algebras of Classical Type

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