Journal of the Mathematical Society of Japan

Isotropy subalgebras of elliptic orbits in semisimple Lie algebras, and the canonical representatives of pseudo-Hermitian symmetric elliptic orbits

Nobutaka BOUMUKI

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Abstract

We give a method of determining the centralizer of an elliptic element in a real semisimple Lie algebra g , in relation with the maximal compact subalgebra of g and the compact dual of g . Moreover, we determine a special central element (called the H -element) of the isotropy subalgebra of each simple irreducible pseudo-Hermitian symmetric Lie algebra.

Article information

Source
J. Math. Soc. Japan, Volume 59, Number 4 (2007), 1135-1177.

Dates
First available in Project Euclid: 10 December 2007

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1197320631

Digital Object Identifier
doi:10.2969/jmsj/05941135

Mathematical Reviews number (MathSciNet)
MR2370009

Zentralblatt MATH identifier
1206.17005

Subjects
Primary: 17B20: Simple, semisimple, reductive (super)algebras
Secondary: 53C35: Symmetric spaces [See also 32M15, 57T15]

Keywords
real semisimple Lie algebra elliptic element centralizer duality pseudo-Hermitian symmetric Lie algebra

Citation

BOUMUKI, Nobutaka. Isotropy subalgebras of elliptic orbits in semisimple Lie algebras, and the canonical representatives of pseudo-Hermitian symmetric elliptic orbits. J. Math. Soc. Japan 59 (2007), no. 4, 1135--1177. doi:10.2969/jmsj/05941135. https://projecteuclid.org/euclid.jmsj/1197320631


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References

  • [1] \auM. Berger, Les espaces symétriques noncompacts, \tiAnn. Sci. École Norm. Sup., , 74 ((1957),)\spg85–\epg177.
  • [2] \auA. Borel and J. de Siebenthal, Les sous-groupes fermés de rang maximum des groupes de Lie clos, \tiComment. Math. Helv., , 23 ((1949),)\spg200–\epg221.
  • [3] \auA. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces, I, \tiAmer. J. Math., , 80 ((1958),)\spg458–\epg538.
  • [4] N. Bourbaki, “Lie groups and Lie algebras, Chapters 4–6”, (originally published as “Groupes et algèbres de Lie,” Hermann, Paris, 1968), Springer-Verlag, Berlin-Heidelberg-New York, 2002.
  • [5] \auJ. Dorfmeister and Z.-D. Guan, Fine structure of reductive pseudo-Kählerian spaces, \tiGeom. Dedicata, , 39 ((1991),)\spg321–\epg338.
  • [6] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, American Mathematical Society, Providence, Rhode Island, 2001.
  • [7] \auS. Kaneyuki, Pseudo-Hermitian symmetric spaces and Siegel domains over nondegenerate cones, \tiHokkaido Math. J., , 20 ((1991),)\spg213–\epg239.
  • [8] S. Kaneyuki, An introduction to affine symmetric spaces, In: the proceedings of the conference “Submanifolds in Yuzawa 2003”, (ed. K. Mashimo), pp. 3–34.
  • [9] T. Kobayashi, Adjoint action, Encyclopaedia of Mathematics, Kluwer Academic Publishers, 1990, pp. 15–16.
  • [10] T. Kobayashi, Harmonic analysis on homogeneous manifolds of reductive type and unitary representation theory, Sūgaku, 46(1994), 124–143; Translations, Series II, Selected Papers on Harmonic Analysis, Groups, and Invariants, (ed. K. Nomizu), 183, Amer. Math. Soc., 1998, pp. 1–31.
  • [11] \auT. Kobayashi and K. Ono, Note on Hirzebruch's proportionality principle, \tiJ. Fac. Sci. Univ. Tokyo Sect. IA, Math., , 37 ((1990),)\spg71–\epg87.
  • [12] \auS. Murakami, Sur la classification des algèbres de Lie réelles et simples, \tiOsaka J. Math., , 2 ((1965),)\spg291–\epg307.
  • [13] I. Satake, Algebraic structures of symmetric domains, Iwanami Shoten, Publishers and Princeton University Press, Tokyo, 1980.
  • [14] \auR. A. Shapiro, Pseudo-Hermitian symmetric spaces, \tiComment. Math. Helv., , 46 ((1971),)\spg529–\epg548.
  • [15] H. Toda and M. Mimura, Topology of Lie groups, I and II, American Mathematical Society, Providence, Rhode Island, 1991.