Singular Localization and Intertwining Functors for Reductive Lie Algebras in Prime Characteristic

By R. Bezrukavnikov, I. Mirkovic, D. Rumynin. Nagoya mathematical journal, 2006

65 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5ef1587182760de8f873306b9daa80d853ece1e8

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Citations in the corpus

Affine braid group actions on derived categories of Springer resolutions (2011, methodology): “(The same proof as in [54] works, taking into account the corrections to [12] given above.”
Braid cobordisms, triangulated categories, and flag varieties (2006, background): “s gives C∗- equivariant kernels generating a braid group action on the derived category of coherent sheaves on the cotangent bundle of the ﬂag variety. 8 Recently Bezrukavnikov, Mirkovi´c and Rumynin [BMR] have also produced an aﬃne braid group action by very diﬀerent and sophisticated ﬁnite charact”