Composantes irréductibles de la variété commutante nilpotente d’une algèbre de Lie symétrique semi-simple  [ Irreducible Components of the Nilpotent Commuting Variety of a Symmetric Semisimple Lie Algebra ]
Annales de l'Institut Fourier, Volume 59 (2009) no. 1, p. 37-80

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.

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.

DOI : https://doi.org/10.5802/aif.2426
Classification:  17B20,  14L30,  17B20
Keywords: Semisimple Lie algebra, symmetric pair, commuting variety, nilpotent orbit
@article{AIF_2009__59_1_37_0,
     author = {Bulois, Micha\"el},
     title = {Composantes irr\'eductibles de la vari\'et\'e commutante nilpotente d'une alg\`ebre de~Lie sym\'etrique semi-simple},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {59},
     number = {1},
     year = {2009},
     pages = {37-80},
     doi = {10.5802/aif.2426},
     mrnumber = {2514861},
     zbl = {1189.17008},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_2009__59_1_37_0}
}
Bulois, Michaël. Composantes irréductibles de la variété commutante nilpotente d’une algèbre de Lie symétrique semi-simple. Annales de l'Institut Fourier, Volume 59 (2009) no. 1, pp. 37-80. doi : 10.5802/aif.2426. http://www.numdam.org/item/AIF_2009__59_1_37_0/

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