Surprising properties of centralisers in classical Lie algebras

Yakimova, Oksana

Annales de l'Institut Fourier, Tome 59 (2009) no. 3, pp. 903-935.

Soit $𝔤$ une algèbre de Lie classique, i.e., ${\mathrm{𝔤𝔩}}_n$, ${\mathrm{𝔰𝔭}}_n$, ou ${\mathrm{𝔰𝔬}}_n$, et soit $e$ un élément nilpotent de $𝔤$. Nous étudions dans cet article diverses propriétés du centralisateur ${𝔤}_e$ de $e$. Les quatre premières sections concernent des problèmes assez élémentaires portant sur le centre de ${𝔤}_e$, la variété commutante de ${𝔤}_e$, ou encore les centralisateurs des paires commutantes. La seconde partie aborde des questions liées aux différentes structures de Poisson sur ${𝔤}_e^{*}$ et aux invariants symétriques de ${𝔤}_e$.

Let $𝔤$ be a classical Lie algebra, i.e., either ${\mathrm{𝔤𝔩}}_n$, ${\mathrm{𝔰𝔭}}_n$, or ${\mathrm{𝔰𝔬}}_n$ and let $e$ be a nilpotent element of $𝔤$. We study various properties of the centralisers ${𝔤}_e$. The first four sections deal with rather elementary questions, like the centre of ${𝔤}_e$, commuting varieties associated with ${𝔤}_e$, or centralisers of commuting pairs. The second half of the paper addresses problems related to different Poisson structures on ${𝔤}_e^{*}$ and symmetric invariants of ${𝔤}_e$.

Classification : 17B45
Keywords: Nilpotent orbits, centralisers, symmetric invariants
Mot clés : orbite nilpotente, centralisateurs, invariants symétriques

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