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), p. 37-80 / Harvested from Numdam

Résumé

Soit $\theta $ une involution de l’algèbre de Lie semi-simple de dimension finie $\mathfrak{g}$ et $\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$ la décomposition de Cartan associée. La variété commutante nilpotente de l’algèbre de Lie symétrique $(\mathfrak{g},\theta )$ est formée des paires d’éléments nilpotents $(x,y)$ de $\mathfrak{p}$ 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 $\mathfrak{p}$-distingués. Cette conjecture a été démontrée par A. Premet dans le cas $(\mathfrak{g}\times \mathfrak{g},\theta )$ avec $\theta (x,y)=(y,x)$. Dans ce travail, nous la prouvons dans un grand nombre d’autres cas.

Abstract

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

Détail

Publié le : 2009-01-01

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

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