Reductive group actions on affine varieties and their doubling

Nous étudions les actions de la forme $(G:X\times X^{*})$ où $X^{*}$ est la $G$-variété duale de $X$. Ces actions sont appelées les doubles. Nous donnons une interprétation géométrique de la complexité de l’action $(G:X)$. Nous montrons que les actions doublées ont un certain nombre de bonnes propriétés, lorsque $X$ est sphérique ou de complexité un.

We study $G$-actions of the form $(G:X\times X^{*})$, where $X^{*}$ is the dual (to $X$) $G$-variety. These actions are called the doubled ones. A geometric interpretation of the complexity of the action $(G:X)$ is given. It is shown that the doubled actions have a number of nice properties, if $X$ is spherical or of complexity one.

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