## Integral homology of loop groups via Langlands dual groups

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- by Zhiwei Yun and Xinwen Zhu
- Represent. Theory
**15**(2011), 347-369 - DOI: https://doi.org/10.1090/S1088-4165-2011-00399-X
- Published electronically: April 20, 2011
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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.## References

- Armand Borel,
*Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts*, Ann. of Math. (2)**57**(1953), 115–207 (French). MR**51508**, DOI 10.2307/1969728 - Armand Borel,
*Commutative subgroups and torsion in compact Lie groups*, Bull. Amer. Math. Soc.**66**(1960), 285–288. MR**0117299**, DOI 10.1090/S0002-9904-1960-10474-0 - Raoul Bott,
*The space of loops on a Lie group*, Michigan Math. J.**5**(1958), 35–61. MR**102803** - Ginzburg, V. Perverse sheaves on a Loop group and Langlands’ duality. arXiv:math/9511007.
- Jens Carsten Jantzen,
*Representations of algebraic groups*, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR**2015057** - Sharad V. Keny,
*Existence of regular nilpotent elements in the Lie algebra of a simple algebraic group in bad characteristics*, J. Algebra**108**(1987), no. 1, 195–201. MR**887203**, DOI 10.1016/0021-8693(87)90133-5 - Bertram Kostant and Shrawan Kumar,
*The nil Hecke ring and cohomology of $G/P$ for a Kac-Moody group $G$*, Adv. in Math.**62**(1986), no. 3, 187–237. MR**866159**, DOI 10.1016/0001-8708(86)90101-5 - Shrawan Kumar, M. S. Narasimhan, and A. Ramanathan,
*Infinite Grassmannians and moduli spaces of $G$-bundles*, Math. Ann.**300**(1994), no. 1, 41–75. MR**1289830**, DOI 10.1007/BF01450475 - George Lusztig,
*Singularities, character formulas, and a $q$-analog of weight multiplicities*, Analysis and topology on singular spaces, II, III (Luminy, 1981) Astérisque, vol. 101, Soc. Math. France, Paris, 1983, pp. 208–229. MR**737932** - Hideyuki Matsumura,
*Commutative algebra*, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR**575344** - I. Mirković and K. Vilonen,
*Geometric Langlands duality and representations of algebraic groups over commutative rings*, Ann. of Math. (2)**166**(2007), no. 1, 95–143. MR**2342692**, DOI 10.4007/annals.2007.166.95 - Bao Châu Ngô,
*Fibration de Hitchin et endoscopie*, Invent. Math.**164**(2006), no. 2, 399–453 (French, with English summary). MR**2218781**, DOI 10.1007/s00222-005-0483-7 - Andrew Pressley and Graeme Segal,
*Loop groups*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1986. Oxford Science Publications. MR**900587** - Neantro Saavedra Rivano,
*Catégories Tannakiennes*, Lecture Notes in Mathematics, Vol. 265, Springer-Verlag, Berlin-New York, 1972 (French). MR**0338002** - Christoph Sorger,
*Lectures on moduli of principal $G$-bundles over algebraic curves*, School on Algebraic Geometry (Trieste, 1999) ICTP Lect. Notes, vol. 1, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2000, pp. 1–57. MR**1795860** - T. A. Springer,
*Some arithmetical results on semi-simple Lie algebras*, Inst. Hautes Études Sci. Publ. Math.**30**(1966), 115–141. MR**206171**, DOI 10.1007/BF02684358

## Bibliographic 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