Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-04T19:10:52.901Z Has data issue: false hasContentIssue false
  • English
  • Français

Equivariant K-theory of compact Lie groups with involution

Published online by Cambridge University Press:  24 February 2014

Po Hu ,
Igor Kriz  and
Petr Somberg
Get access

Abstract

For a compact simply connected simple Lie group G with an involution α, we compute the G ⋊ ℤ/2-equivariant K-theory of G where G acts by conjugation and ℤ/2 acts either by α or by g ↦ α(g)−1. We also give a representation-theoretic interpretation of those groups, as well as of KG(G).

Keywords

equivariant K-theorystable homotopy theorycompact Lie groupscompact symmetric spacesrepresentations of loop groups19L4755N1555P9153C35
Type
Research Article
Information
Journal of K-Theory , Volume 13 , Issue 2 , April 2014 , pp. 313 - 335
Copyright
Copyright © ISOPP 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Adem, A., Gomez, J.M.: Equivariant K-theory of compact Lie group actions with maximal rank isotropy, to appear in J. Topology.Google Scholar
2.Brylinski, J.-L., Zhang, B.: Equivariant K-theory of compact connected Lie groups, K-Theory 20(1) (2000), 23–36.Google Scholar
3.Brylinski, J.-L.: Private correspondence to I. Kriz, 2013.Google Scholar
4.Dold, A., Thom, R.: Quasifaserungen und unendliche symmetrische produkte, Ann. of Math. 67(2) (1958), 239–281.Google Scholar
5.Elmendorf, A., Mandell, M.: Permutative categories as a model of connective stable homotopy, in: Richter, Birgit (ed.), Structured Ring spectra, Cambridge University Press, 2004.Google Scholar
6.Freed, D., Hopkins, M., Teleman, C.: Loop Groups and twisted K-theory I, J. Topology 4 (2011), 737–798.CrossRefGoogle Scholar
7.Frenkel, E.: Lectures on the Langlands Program and Conformal Field Theory, Les Houches 2005.Google Scholar
8.Greenlees, J.P.C., May, J.P.: Completions in algebra and topology, Handbook of Algebraic Topology (ed. James, I. M.), North-Holland, 1995, 255–276.CrossRefGoogle Scholar
9.Greenlees, J.P.C., May, J.P.: Generalized Tate cohomology, Memoirs of the A.M.S. 543 (1995), pp. 178.Google Scholar
10.Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces, Grad. Studies in Math. 34, reprinted with corrections, Amer. Math. Soc. 2001.Google Scholar
11.Hill, M.A., Hopkins, M.J., Ravenel, D.C.: On the non-existence of elements of Kervaire invariant one, Archive preprint: http://arxiv.org/abs/0908.3724.Google Scholar
12.Kriz, I., Lai, L.: On the definition and K-theory realization of a modular functor, preprint, 2013.Google Scholar
13.Lewis, G. Jr., May, J.P., Steinberger, M.: Equivariant stable homotopy theory, Springer, (1986), pp. 538.Google Scholar
14.May, J.P.: A Concise Course in Algebraic Topology, Chicago Lectures in Math. 1999, University of Chicago Press.Google Scholar
15.May, J.P.: Weak equivalences and quasifibrations, Groups of self-equivalences and related topics (Montreal, PQ, 1988), Lecture Notes in Math. 1425, Springer, Berlin (1990) 91–101.Google Scholar
16.Pressley, A., Segal, G.: Loop groups, Oxford Mathematical Monographs, Oxford University Press, New York, 1986.Google Scholar
17.Segal, G.: The representation-ring of a compact Lie group, Publ. Math. I.H.E.S. 34 (1968), 113–128.Google Scholar