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.

Soit θ une involution de l’algèbre de Lie semi-simple de dimension finie 𝔤 et 𝔤=𝔨𝔭 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×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=𝔨𝔭 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×𝔤,θ) where θ(x,y)=(y,x). In this work we prove the conjecture in a significant number of other cases.

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://www.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://www.numdam.org/articles/10.5802/aif.2426/

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