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Semisimple group actions on the three dimensional affine space are linear. (English) Zbl 0645.14020
This paper concerns the linearizability of algebraic group actions on the affine space $${\mathbb{A}}^ n.$$ Kambayashi has conjectured that any action of a linearly reductive group on $${\mathbb{A}}^ n,$$ in any characteristic, is linearizable. The authors show that in characteristic 0, any regular action of a semi-simple algebraic group G on $${\mathbb{A}}^ 3$$ is linearizable. For G of rank 1, the result is proved by using the representation theory of $$SL_ 2.$$
By showing that if every G-invariant function on $${\mathbb{A}}^ n$$is equivalent to a linear action, the result is proved for G of rank $$\geq 2$$ (using the fact that any effective action of G of rank $$\geq 2$$ on $${\mathbb{A}}^ 3$$has a dense orbit). This paper is a nice contribution to geometric invariant theory.
Reviewer: V.Lakshmibai

##### MSC:
 14L30 Group actions on varieties or schemes (quotients) 14L24 Geometric invariant theory 20G05 Representation theory for linear algebraic groups
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