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Integral homology of loop groups via Langlands dual groups


Authors: Zhiwei Yun and Xinwen Zhu
Journal: Represent. Theory 15 (2011), 347-369
MSC (2010): Primary 57T10, 20G07
DOI: https://doi.org/10.1090/S1088-4165-2011-00399-X
Published electronically: April 20, 2011
MathSciNet review: 2788897
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Abstract: Let $ K$ be a connected compact Lie group, and $ G$ its complexification. The homology of the based loop group $ \Omega K$ with integer coefficients is naturally a $ \mathbb{Z}$-Hopf algebra. After possibly inverting $ 2$ or $ 3$, we identify $ H_*(\Omega K,\mathbb{Z})$ with the Hopf algebra of algebraic functions on $ B^\vee_e$, where $ B^\vee$ is a Borel subgroup of the Langlands dual group scheme $ G^\vee$ of $ G$ and $ B^\vee_e$ is the centralizer in $ B^\vee$ of a regular nilpotent element $ e\in\operatorname{Lie} B^\vee$. We also give a similar interpretation for the equivariant homology of $ \Omega K$ under the maximal torus action.


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Additional Information

Zhiwei Yun
Affiliation: Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540
Address at time of publication: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
Email: zyun@math.mit.edu

Xinwen Zhu
Affiliation: Department of Mathematics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02138
Email: xinwenz@math.harvard.edu

DOI: https://doi.org/10.1090/S1088-4165-2011-00399-X
Received by editor(s): September 29, 2009
Received by editor(s) in revised form: October 24, 2010
Published electronically: April 20, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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