## Classification of admissible nilpotent orbits in simple exceptional real Lie algebras of inner type

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- by Alfred G. Noël PDF
- Represent. Theory
**5**(2001), 455-493 Request permission

## Abstract:

In this paper we give a classification of admissible nilpotent orbits of the noncompact simple exceptional real Lie groups of inner type. We use a lemma of Takuya Ohta and some information from the work of Dragomir Djoković to construct a simple algorithm which allows us to decide the admissiblity of a given orbit.## References

- L. Auslander and B. Kostant,
*Polarization and unitary representations of solvable Lie groups*, Invent. Math.**14**(1971), 255–354. MR**293012**, DOI 10.1007/BF01389744
[Bo]Bo N. Bourbaki, - C. W. Curtis,
*Corrections and additions to: “On the degrees and rationality of certain characters of finite Chevalley groups” (Trans. Amer. Math. Soc. 165 (1972), 251–273) by C. T. Benson and Curtis*, Trans. Amer. Math. Soc.**202**(1975), 405–406. MR**364483**, DOI 10.1090/S0002-9947-1975-0364483-8 - Michel Duflo,
*Construction de représentations unitaires d’un groupe de Lie*, Harmonic analysis and group representations, Liguori, Naples, 1982, pp. 129–221 (French, with English summary). MR**777341** - S. Minakshi Sundaram,
*On non-linear partial differential equations of the hyperbolic type*, Proc. Indian Acad. Sci., Sect. A.**9**(1939), 495–503. MR**0000089**, DOI 10.1007/BF03046994 - Dragomir Ž. Đoković,
*Proof of a conjecture of Kostant*, Trans. Amer. Math. Soc.**302**(1987), no. 2, 577–585. MR**891636**, DOI 10.1090/S0002-9947-1987-0891636-0
[Dy]Dy E. Dynkin, - A. A. Kirillov,
*Unitary representations of nilpotent Lie groups*, Uspehi Mat. Nauk**17**(1962), no. 4 (106), 57–110 (Russian). MR**0142001**, DOI 10.1070/RM1962v017n04ABEH004118 - 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 - Anthony W. Knapp,
*Lie groups beyond an introduction*, Progress in Mathematics, vol. 140, Birkhäuser Boston, Inc., Boston, MA, 1996. MR**1399083**, DOI 10.1007/978-1-4757-2453-0
[Ne]Ne M. Nevins, - George Lusztig,
*Some problems in the representation theory of finite Chevalley groups*, The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979) Proc. Sympos. Pure Math., vol. 37, Amer. Math. Soc., Providence, R.I., 1980, pp. 313–317. MR**604598**, DOI 10.1090/pspum/037/604598 - Alfred G. Noël,
*Nilpotent orbits and theta-stable parabolic subalgebras*, Represent. Theory**2**(1998), 1–32. MR**1600330**, DOI 10.1090/S1088-4165-98-00038-7
[No2]No2 A. G. Noël, - Takuya Ohta,
*Classification of admissible nilpotent orbits in the classical real Lie algebras*, J. Algebra**136**(1991), no. 2, 290–333. MR**1089302**, DOI 10.1016/0021-8693(91)90049-E - Jir\B{o} Sekiguchi,
*Remarks on real nilpotent orbits of a symmetric pair*, J. Math. Soc. Japan**39**(1987), no. 1, 127–138. MR**867991**, DOI 10.2969/jmsj/03910127
[Se]Se J. Sekiguchi, - 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**, DOI 10.1007/BFb0081546 - David A. Vogan Jr.,
*Unitary representations of reductive Lie groups*, Annals of Mathematics Studies, vol. 118, Princeton University Press, Princeton, NJ, 1987. MR**908078**, DOI 10.1515/9781400882380 - David A. Vogan Jr.,
*Associated varieties and unipotent representations*, Harmonic analysis on reductive groups (Brunswick, ME, 1989) Progr. Math., vol. 101, Birkhäuser Boston, Boston, MA, 1991, pp. 315–388. MR**1168491**, DOI 10.1007/978-1-4612-0455-8_{1}7

*Groupes et Algèbre de Lie, Chapitres 4,5,6*, Elements de mathématique, Masson (1981).

*Semisimple subalgebras of simple Lie algebras*, Selected papers of E.B. Dynkin with commentary edited by Yushkevich, Seitz and Onishchik AMS, (2000), 175-309. [Dy1]Dy1 E. Dynkin,

*Semisimple subalgebras of simple Lie algebras*, Amer. Soc. Transl. Ser. 2

**6**, (1957), 111-245.

*Admissible nilpotent coadjoint orbits in the p-adic reductive groups*, Ph.D. Thesis M.I.T. Cambridge, MA (June 1998).

*Classification of admissible nilpotent orbits in simple real Lie algebras $E_{6(6)}$ and $E_{6(-26)}$*, Amer. Math. Soc., J. Representation Theory, to appear.

*Remarks on real nilpotent orbits of a symmetric pair*, J. Math. Soc. Japan

**39**, No. 1 (1987), 127-138.

## Additional Information

**Alfred G. Noël**- Affiliation: Department of Mathematics, University of Massachusetts, Boston, Massachusetts 02125
- Address at time of publication: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
- Email: alfred.noel@umb.edu
- Received by editor(s): April 5, 2001
- Received by editor(s) in revised form: September 28, 2001
- Published electronically: November 9, 2001
- © Copyright 2001 American Mathematical Society
- Journal: Represent. Theory
**5**(2001), 455-493 - MSC (2000): Primary 17B20, 17B70
- DOI: https://doi.org/10.1090/S1088-4165-01-00141-8
- MathSciNet review: 1870598