## Integral homology of loop groups via Langlands dual groups

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- by Zhiwei Yun and Xinwen Zhu PDF
- Represent. Theory
**15**(2011), 347-369 Request permission

## 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.## References

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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
- MR Author ID: 862829
- Email: zyun@math.mit.edu
**Xinwen Zhu**- Affiliation: Department of Mathematics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02138
- MR Author ID: 868127
- Email: xinwenz@math.harvard.edu
- Received by editor(s): September 29, 2009
- Received by editor(s) in revised form: October 24, 2010
- Published electronically: April 20, 2011
- © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Represent. Theory
**15**(2011), 347-369 - MSC (2010): Primary 57T10, 20G07
- DOI: https://doi.org/10.1090/S1088-4165-2011-00399-X
- MathSciNet review: 2788897