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Representation Theory

Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.71.

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On Koszul duality for Kac-Moody groups
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by Roman Bezrukavnikov and Zhiwei Yun
Represent. Theory 17 (2013), 1-98
DOI: https://doi.org/10.1090/S1088-4165-2013-00421-1
Published electronically: January 2, 2013

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].
References
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Bibliographic 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