Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 022, 26 pages      arXiv:1308.3871      https://doi.org/10.3842/SIGMA.2014.022

The Real $K$-Theory of Compact Lie Groups

Chi-Kwong Fok
Department of Mathematics, Cornell University, Ithaca, NY 14853, USA

Received August 22, 2013, in final form March 06, 2014; Published online March 11, 2014

Abstract
Let $G$ be a compact, connected, and simply-connected Lie group, equipped with a Lie group involution $\sigma_G$ and viewed as a $G$-space with the conjugation action. In this paper, we present a description of the ring structure of the (equivariant) $KR$-theory of $(G, \sigma_G)$ by drawing on previous results on the module structure of the $KR$-theory and the ring structure of the equivariant $K$-theory.

Key words: $KR$-theory; compact Lie groups; Real representations; Real equivariant formality.

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References

  1. Atiyah M.F., On the $K$-theory of compact Lie groups, Topology 4 (1965), 95-99.
  2. Atiyah M.F., $K$-theory and reality, Quart. J. Math. Oxford Ser. (2) 17 (1966), 367-386.
  3. Atiyah M.F., $K$-theory, W.A. Benjamin, Inc., New York - Amsterdam, 1967.
  4. Atiyah M.F., Bott periodicity and the index of elliptic operators, Quart. J. Math. Oxford Ser. (2) 19 (1968), 113-140.
  5. Atiyah M.F., Segal G.B., Equivariant $K$-theory and completion, J. Differential Geometry 3 (1969), 1-18.
  6. Bourbaki N., Lie groups and Lie algebras, Chapters 7-9, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2005.
  7. Bousfield A.K., On the 2-adic $K$-localizations of $H$-spaces, Homology, Homotopy Appl. 9 (2007), 331-366.
  8. Bröcker T., tom Dieck T., Representations of compact Lie groups, Graduate Texts in Mathematics, Vol. 98, Springer-Verlag, New York, 1985.
  9. Brylinski J.L., Zhang B., Equivariant $K$-theory of compact connected Lie groups, K-Theory 20 (2000), 23-36, dg-ga/9710035.
  10. Fulton W., Harris J., Representation theory, Graduate Texts in Mathematics, Vol. 129, Springer-Verlag, New York, 1991.
  11. Harada M., Landweber G.D., Surjectivity for Hamiltonian $G$-spaces in $K$-theory, Trans. Amer. Math. Soc. 359 (2007), 6001-6025, math.SG/0503609.
  12. Harada M., Landweber G.D., Sjamaar R., Divided differences and the Weyl character formula in equivariant $K$-theory, Math. Res. Lett. 17 (2010), 507-527, arXiv:0906.1629.
  13. Hodgkin L., On the $K$-theory of Lie groups, Topology 6 (1967), 1-36.
  14. Pittie H.V., Homogeneous vector bundles on homogeneous spaces, Topology 11 (1972), 199-203.
  15. Seymour R.M., The real $K$-theory of Lie groups and homogeneous spaces, Quart. J. Math. Oxford Ser. (2) 24 (1973), 7-30.

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