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Representation Theory

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Wall-crossing functors and ${\mathcal{D}}$-modules


Authors: Alexander Beilinson and Victor Ginzburg
Journal: Represent. Theory 3 (1999), 1-31
MSC (1991): Primary 05E99, 17B37
Published electronically: January 11, 1999
MathSciNet review: 1659527
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Abstract: We study Translation functors and Wall-Crossing functors on infinite dimensional representations of a complex semisimple Lie algebra using ${\mathcal{D}}$-modules. This functorial machinery is then used to prove the Endomorphism-theorem and the Structure-theorem; two important results were established earlier by W. Soergel in a totally different way. Other applications to the category ${\mathcal{O}}$ of Bernstein-Gelfand-Gelfand are given, and some conjectural relationships between Koszul duality, Verdier duality and convolution functors are discussed. A geometric interpretation of tilting modules is given.


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Additional Information

Alexander Beilinson
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Email: sasha@math.uchicago.edu

Victor Ginzburg
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Email: ginzburg@math.uchicago.edu

DOI: https://doi.org/10.1090/S1088-4165-99-00063-1
Published electronically: January 11, 1999
Article copyright: © Copyright 1999 American Mathematical Society