2006/02/04 by Roman Bezrukavnikov, Bezrukavnikov, Roman, Ivan Mirković +3
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.math/0602075
openalex publication_date 2006/02/04 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
In math.RT/0205144 we observed that, on the level of derived categories,\nrepresentations of the Lie algebra of a semisimple algebraic group over a field\nof finite characteristic with a given (generalized) regular central character\ncan be identified with coherent sheaves on the formal neighborhood of the\ncorresponding (generalized) Springer fiber. In the present paper we treat\nsingular central characters.\n The basic step is the Beilinson-Bernstein localization of modules with a\nfixed (generalized) central character as sheaves on the partial flag variety\ncorresponding to the singularity of the character. These sheaves are modules\nover a sheaf of algebras which is a version of twisted crystalline differential\noperators, but is actually larger. We discuss translation functors and\nintertwining functors. The latter generate an action of the affine braid group\non the derived category of modules with a regular (generalized) central\ncharacter, which intertwines different localization functors.\n We also describe the standard duality on Lie algebra modules in terms of\nD-modules and coherent sheaves.\n