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Reductive group actions on affine varieties and their doubling. (English) Zbl 0831.14022
Summary: We study $$G$$-actions of the form $$(G:X\times X^{*})$$, where $$X^{*}$$ is the $$G$$-variety dual to $$X$$. These actions are called 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.

##### MSC:
 14L30 Group actions on varieties or schemes (quotients) 13A50 Actions of groups on commutative rings; invariant theory 14L24 Geometric invariant theory
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