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, pp. 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: 10.5802/aif.2426
Classification: 17B20, 14L30, 17B20
Mot clés : algèbre de Lie semi-simple, paire symétrique, variété commutante, orbite nilpotente
Keywords: Semisimple Lie algebra, symmetric pair, commuting variety, nilpotent orbit
Bulois, Michaël 1

1 Université de Brest Département de mathématiques 29238 Brest cedex 3 (France)
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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://archive.numdam.org/articles/10.5802/aif.2426/

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