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Orbits in a real reductive Lie algebra


Author: L. Preiss Rothschild
Journal: Trans. Amer. Math. Soc. 168 (1972), 403-421
MSC: Primary 17B20; Secondary 57E25
DOI: https://doi.org/10.1090/S0002-9947-1972-0349778-3
MathSciNet review: 0349778
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Abstract: The purpose of this paper is to give a classification of the orbits in a real reductive Lie algebra under the adjoint action of a corresponding connected Lie group. The classification is obtained by examining the intersection of the Lie algebra with the orbits in its complexification. An algebraic characterization of the minimal points in the closed orbits is also given.


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  • [1] A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485-535. MR 26 #5081. MR 0147566 (26:5081)
  • [2] E. B. Dynkin, Regular semisimple subalgebras of semisimple Lie algebras, Dokl. Akad. Nauk SSSR 73 (1950), 877-880. (Russian) MR 12, 238. MR 0037291 (12:238g)
  • [3] S. Helgason, Differential geometry and symmetric spaces, Pure and Appl. Math., vol. 12, Academic Press, New York, 1962. MR 26 #2986. MR 0145455 (26:2986)
  • [4] N. Jacobson, Lie algebras, Interscience Tracts in Pure and Appl. Math., no. 10, Interscience, New York, 1962. MR 26 #1345. MR 0143793 (26:1345)
  • [5] B. Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327-404. MR 28 #1252. MR 0158024 (28:1252)
  • [6] B. Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959), 973-1032. MR 22 #5693. MR 0114875 (22:5693)
  • [7] -, On the conjugacy of real Cartan subalgebras. I, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 967-970. MR 17, 509. MR 0073928 (17:509c)
  • [8] B. Kostant and S. Rallis, On orbits associated with symmetric spaces, Bull. Amer. Math. Soc. 75 (1969), 879-883. MR 41 #1935. MR 0257284 (41:1935)
  • [9] -, On representations associated with symmetric spaces, Bull. Amer. Math. Soc. 75 (1969), 884-888. MR 41 #1936. MR 0257285 (41:1936)
  • [10] H. Matsumoto, Quelques remarques sur les groupes de Lie algébriques réels, J. Math. Soc. Japan 16 (1964), 419-446. MR 32 #1292. MR 0183816 (32:1292)
  • [11] L. P. Rothschild, Invariant polynomials and conjugacy classes of real Cartan subalgebras, Indiana Univ. Math. J. 21 (1971). MR 0349777 (50:2270)
  • [12] T. A. Springer, Some arithmetical results on semi-simple Lie algebras, Inst. Hautes Études Sci. Publ. Math. No. 30 (1966), 115-141. MR 34 #5993. MR 0206171 (34:5993)
  • [13] T. A. Springer and R. Steinberg, Conjugacy classes, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N. J., 1968/69), Lecture Notes in Math., vol. 131, Springer, Berlin, 1970, pp. 167-266. MR 42 #3091. MR 0268192 (42:3091)
  • [14] R. Steinberg, Regular elements of semisimple algebraic groups, Inst. Hautes Études Sci. Publ. Math. No. 25 (1965), 49-80. MR 31 #4788. MR 0180554 (31:4788)
  • [15] M. Sugiura, Conjugate classes of Cartan subalgebras in real semisimple Lie algebras, J. Math. Soc. Japan 11 (1959), 374-434. MR 26 #3827. MR 0146305 (26:3827)
  • [16] J. A. Wolf, The action of a real semisimple group on a complex flag manifold. I: Orbit structure and holomorphic arc components, Bull. Amer. Math. Soc. 75 (1969), 1121-1237. MR 40 #4477. MR 0251246 (40:4477)
  • [17] B. Kostant and S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753-809. MR 0311837 (47:399)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0349778-3
Keywords: Real Lie algebra, reductive algebraic group, conjugacy classes, real Lie groups, homogeneous spaces of Lie groups
Article copyright: © Copyright 1972 American Mathematical Society

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