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.

Soit $\theta$ 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 $(𝔤,\theta )$ 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 $(𝔤\times 𝔤,\theta )$ avec $\theta (x,y)=(y,x)$. Dans ce travail, nous la prouvons dans un grand nombre d’autres cas.

Let $\theta$ 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 $(𝔤,\theta )$ 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 $(𝔤\times 𝔤,\theta )$ where $\theta (x,y)=(y,x)$. In this work we prove the conjecture in a significant number of other cases.

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

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