On spherical nilpotent orbits and beyond
Annales de l'Institut Fourier, Tome 49 (1999) no. 5, pp. 1453-1476.

Nous continuons nos investigations concernant la complexité des orbites nilpotentes dans une algèbre de Lie semi-simple. Nous donnons une caractérisation des orbites nilpotentes sphériques au moyen d’une sous-algèbre de Levi minimale qui les rencontre. Ceci fournit une sorte de forme canonique pour ces orbites. Nous obtenons une description des orbites minimales sphériques pour toutes les algèbres de Lie simples. La théorie obtenue pour la représentation adjointe s’étend aux θ-groupes de Vinberg. Nous en déduisons une description des orbites nilpotentes sphériques pour la représentation associée à un espace symétrique.

We continue investigations that are concerned with the complexity of nilpotent orbits in semisimple Lie algebras. We give a characterization of the spherical nilpotent orbits in terms of minimal Levi subalgebras intersecting them. This provides a kind of canonical form for such orbits. A description minimal non-spherical orbits in all simple Lie algebras is obtained. The theory developed for the adjoint representation is then extended to Vinberg’s θ-groups. This yields a description of spherical nilpotent orbits for the isotropy representation of a symmetric variety.

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Panyushev, Dmitri I. On spherical nilpotent orbits and beyond. Annales de l'Institut Fourier, Tome 49 (1999) no. 5, pp. 1453-1476. doi : 10.5802/aif.1726. http://archive.numdam.org/articles/10.5802/aif.1726/

[An82] L.V. Antonyan, On classification of homogeneous elements of ℤ2-graded semisimple Lie algebras, Vestnik Mosk. Un-ta, Ser. Matem. & Mech. No. 2 (1982), 29-34 (Russian). English translation: Moscow Univ. Math. Bulletin, 37, No. 2 (1982), 36-43. | Zbl

[BC76] P. Bala, R.W. Carter, Classes of unipotent elements in simple algebraic groups, II, Math. Proc. Cambridge Philos. Soc., 80 (1976), 1-18. | MR | Zbl

[CM93] D.H. Collingwood, W.M. Mcgovern, Nilpotent orbits in semisimple Lie algebras, New York: Van Nostrand Reinhold, 1993. | MR | Zbl

[Dj88] D. Djoković, Classification of nilpotent elements in simple exceptional real Lie algebras of inner type and description of their centralizers, J. Alg., 112 (1988), 503-524. | MR | Zbl

[DP65] N.M. Dobrovolskaya, V.A. Ponomarev, Pairs of counter operators, Uspekhi Matem. Nauk, 20, No. 6 (1965), 81-86 (Russian). | Zbl

[Dy52] E.B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Matem. Sbornik, 30, No. 2 (1952), 349-462 (Russian). English translation: Amer. Math. Soc. Transl. II, Ser., 6 (1957), 111-244. | MR | Zbl

[El75] A.G. Elashvili, The centralizers of nilpotent elements in semisimple Lie algebras, Trudy Tbiliss. Matem. Inst. Akad. Nauk Gruzin. SSR, 46 (1975), 109-132 (Russian).

[El85] A.G. Elashvili, Frobenius Lie algebras II, Trudy Tbiliss. Matem. Inst. Akad. Nauk Gruzin. SSR, 77 (1985), 127-137 (Russian). | MR | Zbl

[FS97] C.K. Fan, J.R. Stembridge, Nilpotent orbits and commutative elements, J. Algebra, 196 (1997), 490-498. | MR | Zbl

[Ka80] V.G. Kac, Some remarks on nilpotent orbits, J. Algebra, 64 (1980), 190-213. | MR | Zbl

[KR71] B. Kostant, S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math., 93 (1971), 753-809. | MR | Zbl

[KP79] H. Kraft, C. Procesi, Closures of conjugacy classes of matrices are normal, Invent. Math., 53 (1979), 227-247. | MR | Zbl

[Lu72] D. Luna, Sur les orbites fermées des groups algèbriques réductifs, Invent. Math., 16 (1972), 1-5. | MR | Zbl

[Pa87] D. Panyushev, Orbits of maximal dimension of solvable subgroups of reductive algebraic groups and reduction for U-invariants, Matem. Sb., 132, No. 3 (1987), 371-382 (Russian). English translation: Math. USSR-Sb., 60 (1988), 365-375. | Zbl

[Pa94] D. Panyushev, Complexity and nilpotent orbits, Manuscripta Math., 83 (1994), 223-237. | MR | Zbl

[Spal] N. Spaltenstein, “Classes Unipotentes et Sous-groups de Borel”, Lecture notes in Math., 946, Berlin Heidelberg New York: Springer 1982. | MR | Zbl

[Sp74] T.A. Springer, Regular elements in finite reflection groups, Invent. Math., 25 (1974), 159-198. | MR | Zbl

[SpSt] T.A. Springer, R. Steinberg, Conjugacy classes, In: “Seminar on algebraic groups and related finite groups”. Lecture notes in Math., 131, pp. 167-266, Berlin-Heidelberg-New York, Springer, 1970. | MR | Zbl

[Tr83] V.V. Trofimov, Semi-invariants of the coadjoint representation of Borel sub-algebras of simple Lie algebras, In: “Trudy seminara po vect. i tenz. analizu”, vol. 21, pp. 84-105. Moscow: MGU 1983 (Russian). English translation: Selecta Math. Sovietica, 8 (1989), 31-56. | MR | Zbl

[Vi75] E.B. Vinberg, On the classification of nilpotent elements of graded Lie algebras, Dokl. Akad. Nauk SSSR, 225 (1975), No. 4, 745-748 (Russian). English translation: Soviet Math. Dokl., 16 (1975), 1517-1520. | MR | Zbl

[Vi76] E.B. Vinberg, The Weyl group of a graded Lie algebra, Izv. Akad. Nauk SSSR, Ser. Mat., 40 (1976), No. 3, 488-526 (Russian). English translation: Math USSR-Izv., 10 (1976), 463-495. | Zbl

[Vi79] E.B. Vinberg, Classification of homogeneous nilpotent elements of a semisimple graded Lie algebra, In: “Trudy seminara po vect. i tenz. analizu”, vol. 19, pp. 155-177. Moscow: MGU 1979 (Russian). English translation: Selecta Math. Sovietica, 6 (1987), 15-35. | MR | Zbl

[Vi86] E.B. Vinberg, Complexity of actions of reductive groups, Funkt. Anal. i Prilozhen, 20, No. 1 (1986), 1-13 (Russian). English translation: Funct. Anal. Appl., 20 (1986), 1-11. | MR | Zbl

[VP89] E.B. Vinberg, V.L. Popov, Invariant theory, In: Sovremennye problemy matematiki. Fundamentalnye napravleniya, t. 55, pp. 137-309. Moscow: VINITI 1989 (Russian). English translation in: Algebraic Geometry IV (Encyclopaedia Math. Sci., vol. 55, pp. 123-284) Berlin-Heidelberg-New York, Springer, 1994. | Zbl

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