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On spherical nilpotent orbits and beyond. (English) Zbl 0944.17013

Let \({\mathfrak g}\) be a semisimple Lie algebra over an algebraically closed field of characteristic zero. Write \(G\) for the corresponding adjoint group. This paper studies the complexity of nilpotent \(G\)-orbits \({\mathcal O}\) in \({\mathfrak g}\). Here the complexity is defined to be the minimal codimension of a Borel suborbit of \({\mathcal O}\). A nilpotent orbit, \({\mathcal O}\), of complexity zero is called spherical and the author has previously shown this notion is equivalent to the statement \(ht({\mathcal O}) \leq 3\). Here the height of \({\mathcal O}\) is defined as \(ht({\mathcal O})=\max\{n\in{\mathbb N} \mid(ad e)^n\not=0,\) \( e\in{\mathcal O}\}\).
This paper gives an alternate characterization of spherical nilpotent orbits and provides a uniform proof. Namely, \({\mathcal O}\) is spherical if and only if it has a representative which is a sum of simple orthogonal root vectors. Additionally, the author shows the minimal non-spherical orbits in the simple Lie algebras are the ones of complexity 1 for \(SL_n\) and of complexity 2 for the remaining simple Lie algebras. The paper concludes with a study of the complexity nilpotent orbits of \(\theta\)-groups. \(\theta\)-groups are associated with finite order automorphisms, \(\theta\), of \({\mathfrak g}\).

MSC:

17B45 Lie algebras of linear algebraic groups
14L30 Group actions on varieties or schemes (quotients)
14M17 Homogeneous spaces and generalizations
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[1] [An82] , 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 0494.17008
[2] [BC76] , , Classes of unipotent elements in simple algebraic groups, II, Math. Proc. Cambridge Philos. Soc., 80 (1976), 1-18. · Zbl 0364.22007
[3] [CM93] , , Nilpotent orbits in semisimple Lie algebras, New York: Van Nostrand Reinhold, 1993. · Zbl 0972.17008
[4] [Dj88] , Classification of nilpotent elements in simple exceptional real Lie algebras of inner type and description of their centralizers, J. Alg., 112 (1988), 503-524. · Zbl 0639.17005
[5] [DP65] , , Pairs of counter operators, Uspekhi Matem. Nauk, 20, No. 6 (1965), 81-86 (Russian). · Zbl 0161.02801
[6] [Dy52] , 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. · Zbl 0077.03404
[7] [El75] , The centralizers of nilpotent elements in semisimple Lie algebras, Trudy Tbiliss. Matem. Inst. Akad. Nauk Gruzin. SSR, 46 (1975), 109-132 (Russian). · Zbl 0323.17004
[8] [El85] , Frobenius Lie algebras II, Trudy Tbiliss. Matem. Inst. Akad. Nauk Gruzin. SSR, 77 (1985), 127-137 (Russian). · Zbl 0626.17007
[9] [FS97] , , Nilpotent orbits and commutative elements, J. Algebra, 196 (1997), 490-498. · Zbl 0915.20019
[10] [Ka80] , Some remarks on nilpotent orbits, J. Algebra, 64 (1980), 190-213. · Zbl 0431.17007
[11] [KR71] , , Orbits and representations associated with symmetric spaces, Amer. J. Math., 93 (1971), 753-809. · Zbl 0224.22013
[12] [KP79] , , Closures of conjugacy classes of matrices are normal, Invent. Math., 53 (1979), 227-247. · Zbl 0434.14026
[13] [Lu72] , Sur les orbites fermées des groups algèbriques réductifs, Invent. Math., 16 (1972), 1-5. · Zbl 0249.14016
[14] [Pa87] , 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 0663.20044
[15] [Pa94] , Complexity and nilpotent orbits, Manuscripta Math., 83 (1994), 223-237. · Zbl 0822.14024
[16] [Spal] , “Classes Unipotentes et Sous-groups de Borel”, Lecture notes in Math., 946, Berlin Heidelberg New York: Springer 1982. · Zbl 0486.20025
[17] [Sp74] , Regular elements in finite reflection groups, Invent. Math., 25 (1974), 159-198. · Zbl 0287.20043
[18] [SpSt] , , 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. · Zbl 0249.20024
[19] [Tr83] , 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. · Zbl 0659.17009
[20] [Vi75] , 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. · Zbl 0374.17001
[21] [Vi76] , 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 0371.20041
[22] [Vi79] , 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. · Zbl 0431.17006
[23] [Vi86] , 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. · Zbl 0601.14038
[24] [VP89] , , 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 0789.14008
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