Linearizing certain reductive group actions

Abstract

Is every algebraic action of a reductive algebraic group G G on affine space C n {{\mathbf {C}}^n} equivalent to a linear action? The "normal linearization theorem" proved below implies that, if each closed orbit of G G is a fixed point, then C n {{\mathbf {C}}^n} is G G -equivariantly isomorphic to ( C n ) G × C m {({{\mathbf {C}}^n})^G} \times {{\mathbf {C}}^m} for some linear action of G G on C m {{\mathbf {C}}^m} .