On deformation method in invariant theory

Panyushev, Dmitri

Annales de l'Institut Fourier, Tome 47 (1997) no. 4, pp. 985-1012.

Dans cet article nous relions la méthode de déformation en théorie des invariants aux sous-groupes sphériques. Soient $G$ un groupe réductif, $Z$ une $G$-variété affine et $H\subset G$ un sous-groupe sphérique. Lorsque $G/H$ est quasi-affine et que son semi-groupe des poids est saturé, nous montrons que l’algèbre des fonctions régulières $H$-invariantes sur $Z$ a une filtration stable par $G$ telle que l’algèbre graduée associée est l’algèbre des invariants d’un sous-groupe horosphérique explicite de $G$. La méthode de déformation sous sa forme habituelle, développée par Luna et d’autres auteurs, est un cas particulier de cette construction. Notre résultat s’applique aussi à la description des invariants de certaines représentations réductibles des groupes réductifs.

Nous donnons ensuite de nouvelles applications de la méthode de déformation ; elles concernent les algèbres d’invariants qui sont intersection complète et aussi les actions doublées.

In this paper we relate the deformation method in invariant theory to spherical subgroups. Let $G$ be a reductive group, $Z$ an affine $G$-variety and $H\subset G$ a spherical subgroup. We show that whenever $G/H$ is affine and its semigroup of weights is saturated, the algebra of $H$-invariant regular functions on $Z$ has a $G$-invariant filtration such that the associated graded algebra is the algebra of regular functions of some explicit horospherical subgroup of $G$. The deformation method in its usual form, as developed by Luna et al., is a particular case of this construction. Our result also applies to the description of invariants of some reducible representations of reductive groups.

New applications of the deformation method are given which concern the property of being complete intersection for algebras of invariants. We also give some applications of the deformation method to doubled actions.

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