On polynomial automorphisms of affine spaces

Abstract

In the first part of this paper we prove some general results on the linearizability of algebraic group actions on . As an application, we get a method of construction and concrete examples of non-linearizable algebraic actions of infinite non-reductive insoluble algebraic groups on with a fixed point. In the second part we use these general results to prove that every effective algebraic action of a connected reductive algebraic group on the -dimensional affine space over an algebraically closed field of characteristic zero is linearizable in each of the following cases: 1) ; 2) and is not a one- or two-dimensional torus. In particular, this means that is the unique (up to conjugacy) maximal connected reductive subgroup of the automorphism group of the algebra of polynomials in three variables over .