## Orbits in a real reductive Lie algebra

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- by L. Preiss Rothschild PDF
- Trans. Amer. Math. Soc.
**168**(1972), 403-421 Request permission

## 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

## Additional Information

- © Copyright 1972 American Mathematical Society
- 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