Convexity and Loop Groups

  • Michael Francis Atiyah
  • Andrew Nicholas Pressley
Part of the Progress in Mathematics book series (PM, volume 36)


The purpose of this paper is to extend certain convexity results associated with compact Lie groups to an infinite-dimensional setting, in which the Lie group is replaced by the corresponding loop group. To recall the finite-dimensional results which we shall generalize let G be a simply connected, compact Lie group, T a maximal torus of G and W its Weyl group. Consider the adjoint action of G on its Lie algebra L(G) and fix a G-invariant metric on L(C) so that we can define orthogonal projection. A result of Kostant [8] describes the images of the G-orbits in L(G) under the orthogonal projection onto L(T). To state it, recall that such G-orbits correspond to W-orbits in L(T). Then Kostant's result is:

(1.1). The orthogonal projection of a G-orbit onto L(T) coincides with the convex hull of the corresponding W-orbit.


Convex Hull Weyl Group Symplectic Form Maximal Torus Loop Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), 1 - 15.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    N. Bourbaki, Groupes et algèbres de Lie, Ch. 4-6, Hermann (Paris) 1968.Google Scholar
  3. [3]
    V. Guillemin & S. Sternberg, Convexity properties of the moment mapping, Invent. Math. 67, 491 - 513 (1982).MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    G. Heckman, Thesis, Leiden (1980).Google Scholar
  5. [5]
    N. Iwahori & H. Matsumoto, On some Bruhat decompositions and the structure of Hecke rings of p-adic Chevalley groups, Publ. Math. I.II.E.S. (Paris) 25 (1965), 5 - 48.MathSciNetzbMATHGoogle Scholar
  6. [6]
    V. G. Kac, Simple irreducible graded Lie algebras of finite growth, Izv. Akad. Nauk. 32 (1968), 1271 - 1311.Google Scholar
  7. [7]
    A. A. Kirillov, Elements of the theory of representations, Springer-Verlag 1978.Google Scholar
  8. [8]
    B. Kostant, On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. Éc. Norm. Sup. 6 (1973), 413 - 455.MathSciNetzbMATHGoogle Scholar
  9. [9]
    R. S. Palais, Morse theory on Hilbert manifolds, Topology 2 (1962), 299 - 340.MathSciNetCrossRefGoogle Scholar
  10. [10]
    A. N. Pressley, Decompositions of the space of loops on a Lie group, Topology 19 (1980), 65 - 79.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    A. N. Pressley, Thesis, University of Oxford (1980).Google Scholar
  12. [12]
    A. N. Pressley, The energy flow on the loop space of a compact Lie group, J. London Math. Soc. (to appear).Google Scholar
  13. [13]
    G. B. Segal, Unitary representations of some infinite dimensional groups, Commun. Math. Phys. 80 (1981), 301 - 342.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1983

Authors and Affiliations

  • Michael Francis Atiyah
    • 1
  • Andrew Nicholas Pressley
    • 1
  1. 1.Mathematical InstituteOxfordEngland

Personalised recommendations