Orbits in a real reductive Lie algebra

Rothschild, L. Preiss

doi:10.1090/S0002-9947-1972-0349778-3

Orbits in a real reductive Lie algebra
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by L. Preiss Rothschild

Trans. Amer. Math. Soc. 168 (1972), 403-421

DOI: https://doi.org/10.1090/S0002-9947-1972-0349778-3

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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.

References

Armand Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485–535. MR 147566, DOI 10.2307/1970210
E. B. Dynkin, Regular semisimple subalgebras of semisimple Lie algebras, Doklady Akad. Nauk SSSR (N.S.) 73 (1950), 877–880 (Russian). MR 0037291
Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455
Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0143793
Bertram Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404. MR 158024, DOI 10.2307/2373130
Bertram Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959), 973–1032. MR 114875, DOI 10.2307/2372999
Bertram Kostant, On the conjugacy of real Cartan subalgebras. I, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 967–970. MR 73928, DOI 10.1073/pnas.41.11.967
Bertram Kostant and Stephen Rallis, On orbits associated with symmetric spaces, Bull. Amer. Math. Soc. 75 (1969), 879–883. MR 257284, DOI 10.1090/S0002-9904-1969-12337-2
Bertram Kostant and Stephen Rallis, On representations associated with symmetric spaces, Bull. Amer. Math. Soc. 75 (1969), 884–888. MR 257285, DOI 10.1090/S0002-9904-1969-12339-6
Hideya Matsumoto, Quelques remarques sur les groupes de Lie algébriques réels, J. Math. Soc. Japan 16 (1964), 419–446 (French). MR 183816, DOI 10.2969/jmsj/01640419
L. Preiss Rothschild, Invariant polynomials and conjugacy classes of real Cartan subalgebras, Indiana Univ. Math. J. 21 (1971/72), 115–120. MR 349777, DOI 10.1512/iumj.1971.21.21010
T. A. Springer, Some arithmetical results on semi-simple Lie algebras, Inst. Hautes Études Sci. Publ. Math. 30 (1966), 115–141. MR 206171, DOI 10.1007/BF02684358
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 Mathematics, Vol. 131, Springer, Berlin, 1970, pp. 167–266. MR 0268192
Robert Steinberg, Regular elements of semisimple algebraic groups, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 49–80. MR 180554, DOI 10.1007/BF02684397
Mitsuo Sugiura, Conjugate classes of Cartan subalgebras in real semi-simple Lie algebras, J. Math. Soc. Japan 11 (1959), 374–434. MR 146305, DOI 10.2969/jmsj/01140374
Joseph 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 251246, DOI 10.1090/S0002-9904-1969-12359-1
B. Kostant and S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753–809. MR 311837, DOI 10.2307/2373470