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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
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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  • [B53] Borel, A. Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts. Ann. of Math. (2) 57, (1953). 115-207. MR 0051508 (14:490e)
  • [B60] Borel, A. Commutative subgroups and torsion in compact Lie groups. Bull. amer. Math. Soc. 66 1960 285-288. MR 0117299 (22:8080)
  • [B58] Bott, R. The space of loops on a Lie group. Michigan Math. J. 5(1958), 35-61. MR 0102803 (21:1589)
  • [G95] Ginzburg, V. Perverse sheaves on a Loop group and Langlands' duality. arXiv:math/9511007.
  • [J03] Jantzen, J. C. Representations of algebraic groups. Second edition. Mathematical Surveys and Monographs, 107. American Mathematical Society, Providence, RI, 2003. MR 2015057 (2004h:20061)
  • [K87] Keny, S. V. 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 (88d:17011)
  • [KK86] Kostant, B; Kumar, S. The nil Hecke ring and the cohomology of $ G/P$ for a Kac-Moody group $ G$. Adv. in Math. 62 (1986), 187-237. MR 866159 (88b:17025b)
  • [KNR94] Kumar, S; Narasimhan, M.S; Ramanathan, A. Infinite Grassmannians and moduli spaces of $ G$-bundles. Math. Annalen 300 (1994), 41-75. MR 1289830 (96e:14011)
  • [L81] Lusztig, G. Singularities, character formulas, and a $ q$-analog of weight multiplicities. Analysis and topology on singular spaces, II, III (Luminy, 1981), 208-229, Astérisque, 101-102, Soc. Math. France, Paris, 1983. MR 737932 (85m:17005)
  • [M80] Matsumura, H. Commutative algebra. Second edition. Mathematics Lecture Note Series, 56. Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344 (82i:13003)
  • [MV07] Mirković, I.; Vilonen, K. Geometric Langlands duality and representations of algebraic groups over commutative rings. Ann. of Math. (2) 166 (2007), no. 1, 95-143. MR 2342692 (2008m:22027)
  • [N06] Ngô, B-C. Fibration de Hitchin et endoscopie. Invent. Math. 164 (2006), no. 2, 399-453. MR 2218781 (2007k:14018)
  • [PS86] Pressley, A; Segal, G. Loop groups. The Clarendon Press Oxford University Press, New York, 1986. Oxford Science Publications. MR 900587 (88i:22049)
  • [Sa72] Saavedra Rivano, N. Catégories Tannakiennes. Lecture Notes in Mathematics, Vol. 265. Springer-Verlag, Berlin-New York, 1972. MR 0338002 (49:2769)
  • [So00] Sorger, C. Lectures on moduli of principal $ G$-bundles over algebraic curves. School on Algebraic Geometry (Trieste, 1999), 1-57, ICTP Lect. Notes, 1, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2000. MR 1795860 (2002h:14017)
  • [Sp66] Springer, T. A. Some arithmetical results on semi-simple Lie algebras. Inst. Hautes Études Sci. Publ. Math. No. 30 (1966), 115-141. MR 0206171 (34:5993)

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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

Xinwen Zhu
Affiliation: Department of Mathematics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02138

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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