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


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