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

Published online by Cambridge University Press:  11 January 2016

Roman Bezrukavnikov ,
Ivan Mirković  and
Dmitriy Rumynin
Roman Bezrukavnikov
Affiliation:
Department of MathematicsMassachusetts Institute of Technology77 Massachusetts ave. Cambridge, MA 02139U.S.A.bezrukav@math.mit.edu
Ivan Mirković
Affiliation:
Department of Mathematics and StatisticsUniversity of MassachusettsAmherst, MA 01003U.S.A.mirkovic@math.umass.edu
Dmitriy Rumynin
Affiliation:
Mathematics DepartmentUniversity of WarwickCoventry, CV4 7ALEnglandrumynin@maths.warwick.ac.uk
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Abstract

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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.

Keywords

17B5016E2018F99
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 183 , 2006 , pp. 1 - 55
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2006

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