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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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On Koszul duality for Kac-Moody groups
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by Roman Bezrukavnikov and Zhiwei Yun PDF
Represent. Theory 17 (2013), 1-98 Request permission

Abstract:

For any Kac-Moody group $G$ with Borel $B$, we give a monoidal equivalence between the derived category of $B$-equivariant mixed complexes on the flag variety $G/B$ and (a certain completion of) the derived category of $G^\vee$-monodromic mixed complexes on the enhanced flag variety $G^\vee /U^\vee$, here $G^\vee$ is the Langlands dual of $G$. We also prove variants of this equivalence, one of which is the equivalence between the derived category of $U$-equivariant mixed complexes on the partial flag variety $G/P$ and a certain “Whittaker model” category of mixed complexes on $G^\vee /B^\vee$. In all these equivalences, intersection cohomology sheaves correspond to (free-monodromic) tilting sheaves. Our results generalize the Koszul duality patterns for reductive groups in [BGS96].
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Additional Information
  • Roman Bezrukavnikov
  • Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
  • MR Author ID: 347192
  • Email: bezrukav@math.mit.edu
  • Zhiwei Yun
  • Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
  • Address at time of publication: Department of Mathematics, Stanford University, 450 Serra Mall, Stanford, California 94305
  • MR Author ID: 862829
  • Email: zyun@stanford.edu
  • Received by editor(s): January 15, 2011
  • Received by editor(s) in revised form: July 7, 2011, August 13, 2011, and April 11, 2012
  • Published electronically: January 2, 2013
  • Additional Notes: The first author was partly supported by the NSF grant DMS-0854764.
    The second author was supported by the NSF grant DMS-0635607 and Zurich Financial Services as a member at the Institute for Advanced Study, and by the NSF grant DMS-0969470.
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Represent. Theory 17 (2013), 1-98
  • MSC (2010): Primary 20G44, 14M15, 14F05
  • DOI: https://doi.org/10.1090/S1088-4165-2013-00421-1
  • MathSciNet review: 3003920