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Nilpotent pairs, dual pairs, and sheets. (English) Zbl 1136.17303
This paper continues the study begun by V. Ginzburg [Invent. Math. 140, 511–561 (2000; Zbl 0984.17007)] of pairs of commuting nilpotent elements in complex semisimple Lie algebras \(\mathfrak g\). The earlier work had focussed on principal nilpotent pairs, that is, pairs whose simultaneous centralizer has the least possible dimension (equal to the rank of \(\mathfrak g\)). Here the author shows that many of the results in that setting extend to almost principal pairs, that is, pairs whose simultaneous centralizer is one more than the rank of \(\mathfrak g\). Some new phenomena do however occur. For example, the elements in the semisimple pair \((h_1,h_2)\) corresponding to an almost principal pair \((e_1,e_2)\) need not have integral eigenvalues.
The author also develops some beautiful connections between nilpotent pairs and Howe’s reductive dual pairs. Given a nilpotent pair \((e_1,e_2)\) and the corresponding semisimple pair \((h_1,h_2)\), he shows that the centralizers \({\mathfrak z}(e_i,h_i)\) form a dual pair in \(\mathfrak g\) under certain conditions, which are always satisfied if the pair is principal or almost principal. The dual pair is reductive if and only if the original nilpotent pair can be embedded in a direct sum of commuting copies of \(\mathfrak{sl}_2\). A class of particularly nice nilpotent pairs called excellent is defined and classified for the simple Lie algebras. To such pairs one can attach smooth sheets with affine sections, generalizing a well-known construction of Kostant for principal nilpotent orbits in semisimple Lie algebras.

17B20 Simple, semisimple, reductive (super)algebras
17B05 Structure theory for Lie algebras and superalgebras
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[1] Alekseevskiǐ, A.V., The component groups of the centralizers of unipotent elements in semisimple algebraic groups, Trudy razmadze mat. inst. (Tbilisi), 62, 5-27, (1979)
[2] Borho, W.; Kraft, H., Über bahnen und deren deformationen bei aktionen reduktiver gruppen, Comment. math. helv., 54, 61-104, (1979) · Zbl 0395.14013
[3] Collingwood, D.H.; McGovern, W.M., Nilpotent orbits in semisimple Lie algebras, (1993), Van Nostrand-Reinhold New York · Zbl 0972.17008
[4] Dynkin, E.B., Semisimple subalgebras of semisimple Lie algebras, Mat. sb., 30, 349-462, (1952) · Zbl 0048.01701
[5] Elashvili, A.G., The centralizers of nilpotent elements in semisimple Lie algebras, Trudy razmadze mat. inst. (Tbilisi), 46, 109-132, (1975) · Zbl 0323.17004
[6] Elashvili, A.G., Sheets in the exceptional Lie algebras, Issledovaniya po algebre, (1985), p. 171-194
[7] A. G. Elashvili and D. Panyushev, Towards a classification of principal nilpotent pairs, Appendix of Ref. [9]. · Zbl 1011.17009
[8] A. G. Elashvili, and, D. Panyushev, A classification of the principal nilpotent pairs in simple Lie algebras and related problems, J. London Math. Soc, in press. · Zbl 1011.17009
[9] Ginzburg, V., Principal nilpotent pairs in a semisimple Lie algebra, I, Invent. math., 140, 511-561, (2000) · Zbl 0984.17007
[10] Katsylo, P.I., Sections of sheets in a reductive algebraic Lie algebra, Izv. AN SSSR. ser. mat., 46, 477-486, (1982)
[11] Kempken, G., Induced conjugacy classes in classical Lie-algebras, Abh. math. sem. univ. Hamburg, 53, 53-83, (1983) · Zbl 0495.17003
[12] Kostant, B., The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. math., 81, 973-1032, (1959) · Zbl 0099.25603
[13] Kostant, B., Lie group representations in polynomial rings, Amer. J. math., 85, 327-404, (1963) · Zbl 0124.26802
[14] Panyushev, D., On spherical nilpotent orbits and beyond, Ann. inst. Fourier, 49, 1453-1476, (1999) · Zbl 0944.17013
[15] Panyushev, D., Nilpotent pairs in semisimple Lie algebras and their characteristics, Internat. math. res. notices, 1, 1-21, (2000) · Zbl 0954.17007
[16] Richardson, R.W., Commuting varieties of semisimple Lie algebras and algebraic groups, Compositio math., 38, 311-327, (1979) · Zbl 0409.17006
[17] Rubenthaler, H., Paramétrisation d’orbites dans LES nappes de Dixmier admissibles, Mém. soc. math. France, 15, 255-275, (1984) · Zbl 0574.17006
[18] Rubenthaler, H., LES paires duales dans LES algèbres de Lie reductives, Asterisque, 219, (1994) · Zbl 0805.17008
[19] Spaltenstein, N., Classes unipotentes et sous-groupes de Borel, Lecture notes in math., 946, (1982), Springer-Verlag Berlin/Heidelberg/New York · Zbl 0486.20025
[20] Springer, T.A.; Steinberg, R., Conjugacy classes, Seminar on algebraic groups and related finite groups, Lecture notes in math., 131, (1970), Springer-Verlag Berlin/New York, p. 167-266 · Zbl 0249.20024
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