Geodesic
The Index of Centralizers of Elements in Classical Lie Algebras
Funkcionalʹnyj analiz i ego priloženiâ, Tome 40 (2006) no. 1, pp. 52-64.

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The index of a finite-dimensional Lie algebra $\mathfrak{g}$ is the minimum of dimensions of the stabilizers $\mathfrak{g}_\alpha$ over all covectors $\alpha\in\mathfrak{g}^*$. Let $\mathfrak{g}$ be a reductive Lie algebra over a field $\mathbb{K}$ of characteristic $\ne2$. Élashvili conjectured that the index of $\mathfrak{g}_\alpha$ is always equal to the index, or, which is the same, the rank of $\mathfrak{g}$. In this article, Élashvili's conjecture is proved for classical Lie algebras. Furthermore, it is shown that if $\mathfrak{g}=\mathfrak{gl}_n$ or $\mathfrak{g}=\mathfrak{sp}_{2n}$ and $e\in\mathfrak{g}$ is a nilpotent element, then the coadjoint action of $\mathfrak{g}_e$ has a generic stabilizer. For $\mathfrak{g}=\mathfrak{so}_n$, we give examples of nilpotent elements $e\in\mathfrak{g}$ such that the coadjoint action of $\mathfrak{g}_e$ does not have a generic stabilizer. 
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O. S. Yakimova. The Index of Centralizers of Elements in Classical Lie Algebras. Funkcionalʹnyj analiz i ego priloženiâ, Tome 40 (2006) no. 1, pp. 52-64. http://geodesic.mathdoc.fr/item/FAA_2006_40_1_a4/

[1] A. V. Bolsinov, “Kriterii polnoty semeistva funktsii v involyutsii, postroennogo metodom sdviga argumenta”, Dokl. AN SSSR, 301:5 (1988), 1037–1040 | MR

[2] Zh. Diksme, Universalnye obertyvayuschie algebry, Mir, M., 1978 | MR

[3] A. G. Elashvili, “Ob indekse orisfericheskikh podalgebr poluprostykh algebr Li”, Trudy Tbilissk. matem. in-ta im. Razmadze, 77 (1985), 116–126 | MR

[4] A. G. Elashvili, “Kanonicheskii vid i statsionarnye podalgebry tochek obschego polozheniya dlya prostykh lineinykh grupp Li”, Funkts. analiz i ego pril., 6:1 (1972), 51–62 | MR | Zbl

[5] J.-Y. Charbonnel, “Propriétés (Q) and (C). Variété commutante”, Bull. Soc. Math. France, 132:4 (2004), 477–508 | DOI | MR | Zbl

[6] R. M. Guralnick, “A note on commuting pairs of matrices”, Linear and Multilinear Algebra, 31 (1992), 71–75 | DOI | MR | Zbl

[7] M. G. Neubauer, B. A. Sethuraman, “Commuting pairs in the centralizers of 2-regular matrices”, J. Algebra, 214:1 (1999), 174–181 | DOI | MR | Zbl

[8] D. Panyushev, “Inductive formulas for the index of seaweed Lie algebras”, Mosc. Math. J., 1:2 (2001), 221–241 | DOI | MR | Zbl

[9] D. Panyushev, “The index of a Lie algebra, the centralizer of a nilpotent element, and the normalizer of the centralizer”, Math. Proc. Cambr. Phil. Soc., 134:1 (2003), 41–59 | DOI | MR | Zbl

[10] D. Panyushev, “Some amazing properties of spherical nilpotent orbits”, Math. Z., 245:3 (2003), 557–580 | DOI | MR | Zbl

[11] D. Panyushev, “An extension of Raïs' theorem and seaweed subalgebras of simple Lie algebras”, Ann. Inst. Fourier, 55:3 (2005), 693–715 | DOI | MR | Zbl

[12] M. Rosenlicht, “A remark on quotient spaces”, An. Acad. Brasil. Cienc., 35 (1963), 487–489 | MR | Zbl

[13] P. Tauvel,R. W. T. Yu, “Indice et formes linéaires stables dans les algèbres de Lie”, J. Algebra, 273:2 (2004), 507–516 | DOI | MR | Zbl