Composantes irréductibles de la variété commutante nilpotente d’une algèbre de Lie symétrique semi-simple

Bulois, Michaël

doi:10.5802/aif.2426

Composantes irréductibles de la variété commutante nilpotente d’une algèbre de Lie symétrique semi-simple

Bulois, Michaël

Annales de l'Institut Fourier, Tome 59 (2009) no. 1, pp. 37-80.

Résumé
Abstract

Soit $θ$ une involution de l’algèbre de Lie semi-simple de dimension finie $𝔤$ et $𝔤 = 𝔨 \oplus 𝔭$ la décomposition de Cartan associée. La variété commutante nilpotente de l’algèbre de Lie symétrique $(𝔤, θ)$ est formée des paires d’éléments nilpotents $(x, y)$ de $𝔭$ tels que $[x, y] = 0$ . Il est conjecturé que cette variété est équidimensionnelle et que ses composantes irréductibles sont indexées par les orbites d’éléments $𝔭$ -distingués. Cette conjecture a été démontrée par A. Premet dans le cas $(𝔤 \times 𝔤, θ)$ avec $θ (x, y) = (y, x)$ . Dans ce travail, nous la prouvons dans un grand nombre d’autres cas.

Let $θ$ be an involution of the finite dimensional semisimple Lie algebra $𝔤$ and $𝔤 = 𝔨 \oplus 𝔭$ be the associated Cartan decomposition. The nilpotent commuting variety of $(𝔤, θ)$ consists in pairs of nilpotent elements $(x, y)$ of $𝔭$ such that $[x, y] = 0$ . It is conjectured that this variety is equidimensional and that its irreducible components are indexed by the orbits of $𝔭$ distinguished elements. This conjecture was established by A. Premet in the case $(𝔤 \times 𝔤, θ)$ where $θ (x, y) = (y, x)$ . In this work we prove the conjecture in a significant number of other cases.

MR 2514861 | Zbl 1189.17008

DOI : https://doi.org/10.5802/aif.2426
Classification : 17B20, 14L30, 17B20
Mots clés : algèbre de Lie semi-simple, paire symétrique, variété commutante, orbite nilpotente

@article{AIF_2009__59_1_37_0,
     author = {Bulois, Micha\"el},
     title = {Composantes irr\'eductibles de la vari\'et\'e commutante nilpotente d{\textquoteright}une alg\`ebre de~Lie sym\'etrique semi-simple},
     journal = {Annales de l'Institut Fourier},
     pages = {37--80},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
     number = {1},
     year = {2009},
     doi = {10.5802/aif.2426},
     mrnumber = {2514861},
     zbl = {1189.17008},
     language = {fr},
     url = {http://archive.numdam.org/articles/10.5802/aif.2426/}
}

Bulois, Michaël. Composantes irréductibles de la variété commutante nilpotente d’une algèbre de Lie symétrique semi-simple. Annales de l'Institut Fourier, Tome 59 (2009) no. 1, pp. 37-80. doi : 10.5802/aif.2426. http://archive.numdam.org/articles/10.5802/aif.2426/

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