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Representation Theory
Representation Theory
ISSN 1088-4165

On Koszul duality for Kac-Moody groups


Authors: Roman Bezrukavnikov and Zhiwei Yun
Journal: Represent. Theory 17 (2013), 1-98
MSC (2010): Primary 20G44, 14M15, 14F05
Posted: January 2, 2013
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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
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
Email: zyun@stanford.edu

DOI: http://dx.doi.org/10.1090/S1088-4165-2013-00421-1
PII: S 1088-4165(2013)00421-1
Received by editor(s): January 15, 2011
Received by editor(s) in revised form: July 7, 2011, August 13, 2011, and April 11, 2012
Posted: 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.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.