An extension of Rais' theorem and seaweed subalgebras of simple Lie algebras
[Extension d'un théorème de Rais et sous-algèbres de Lie simples]
I. Panyushev, Dmitri1
1 Independent university of Moscow, Bol'shoi Vlaservskii per.11, 119002 Moscow (Russia)
Annales de l'Institut Fourier, Tome 55 (2005) no. 3, pp. 693-715.

Nous obtenons une extension d'un théorème de Rais sur la représentation coadjointe de certaines algèbres de Lie graduées. Comme application, nous démontrons que la représentation coadjointe d'une sous-algèbre spéciale dans une algèbre de Lie simple de type A ou C possède un stabilisateur générique, et que son corps des invariants est rationnel. Nous montrons aussi que si la plus grande racine d'une algèbre de Lie simple n'est pas un poids fondamental, alors il existe une sous-algèbre parabolique dont la représentation coadjointe n'admet pas de stabilisateur générique.

We prove an extension of Rais' theorem on the coadjoint representation of certain graded Lie algebras. As an application, we prove that, for the coadjoint representation of any seaweed subalgebra in a general linear or symplectic Lie algebra, there is a generic stabiliser and the field of invariants is rational. It is also shown that if the highest root of a simple Lie algerba is not fundamental, then there is a parabolic subalgebra whose coadjoint representation do not have a generic stabiliser.

DOI : 10.5802/aif.2110
Classification : 17B20, 17B70, 14L30
Keywords: field of invariants, generic stabiliser, simple Lie algebra, seaweed subalgebra
Mot clés : corps des invariants, stabilisateur générique, algèbre de Lie simple, sous-algèbre spéciale.
I. Panyushev, Dmitri 1

1 Independent university of Moscow, Bol'shoi Vlaservskii per.11, 119002 Moscow (Russia) 
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I. Panyushev, Dmitri. An extension of Rais' theorem and seaweed subalgebras of simple Lie algebras. Annales de l'Institut Fourier, Tome 55 (2005) no. 3, pp. 693-715. doi : 10.5802/aif.2110. http://www.numdam.org/articles/10.5802/aif.2110/

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