On the linearization of actions of linearly reductive groups (Q912174)

Let G be a linearly reductive group acting on an affine space \(A^ n\). Suppose G has a fixed point and acts by polynomial transformations of degree two. If the ground field k has characteristic two, assume moreover that G is abelian. Then it is proved that the action is linearizable. An explicit formula, using the Reynolds operator, is given for an automorphism of \(A^ n\), ``linearizing'' the action. Remark: 1. If char k\(=0\), then the claim has been recently proved under weaker assumption [see the author's paper ``On some reductive group actions on affine space'' in Group actions and invariant theory, Proc. Conf., Montreal/Can. 1988, Conf. Proc. 10, 67-72 (1989). \(-\quad 2.\quad In\) general the action of reductive groups on affine space is not linearizable, see \textit{G. W. Schwarz}, C. R. Acad. Sci. Paris, Sér. I Math. 309, No.2, 89-94 (1989; Zbl 0688.14040)].