Linearizing Certain Reductive Group Actions

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

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