Canonical Forms Under Certain Actions on the Classical Compact Simple Lie Groups

  • Osamu Ikawa
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 106)


A maximal torus of a compact connected Lie group can be seen as a canonical form of adjoint action since any two maximal tori can be transformed each other by an inner automorphism. A. Kollross defined a σ-action on a compact Lie group which is a generalization of the adjoint action. Since a σ-action is hyperpolar, it has a canonical form called a section. In this paper we study the structure of the orbit space of a σ-action and properties of each orbit, such as minimal, austere and totally geodesic, using symmetric triads introduced by the author, when σ is an involution of outer type on the compact simple Lie groups of classical type. As an application, we investigate the fixed point set of a holomorphic isometry of an irreducible Hermitian symmetric space of compact type which does not belong to the identity component of the group of holomorphic isometries.


Maximal Torus Orbit Space Adjoint Action Outer Automorphism Holomorphic Isometry 
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The author was partly supported by the Grant-in-Aid for Science Research (C) 2013 (No. 25400070), Japan Society for the Promotion of Science.


  1. 1.
    Harvey, R., Lawson, Jr. H.B.: Calibrated geometries. Acta Math. 148, 47–157 (1982)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Heintze, E., Palais, R.S., Terng, C.-L., Thorbergsson, G.: Hyperpolar actions on symmetric spaces. Geometry, topology, & physics, pp. 214–245. Conf. Proc. Lecture Notes Geom. Topology, IV. Int. Press, Cambridge, MA (1995)Google Scholar
  3. 3.
    Ikawa, O.: The geometry of symmetric triad and orbit spaces of Hermann actions. J. Math. Soc. Jpn. 63, 70–136 (2011)Google Scholar
  4. 4.
    Ikawa, O., Tanaka, M.S., Tasaki, H.: The fixed point set of a holomorphic isometry, the intersection of two real forms in a Hermitian symmetric space of compact type and symmetric triads. PreprintGoogle Scholar
  5. 5.
    Kollross, A.: A classification of hyperpolar and cohomogeneity one actions. Trans. Am. Math. Soc. 354, 571–612 (2002)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Matsuki, T.: Classification of two involutions on compact semisimple Lie groups and root systems. J. Lie Theory 12, 41–68 (2002)MATHMathSciNetGoogle Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.Faculty of Arts and Sciences, Department of Mathematics and Physical SciencesKyoto Institute of Technology, MatsugasakiSakyoku, KyotoJapan

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