Abstract
Let G subset of GL(V) be a complex reductive group where dim V < infinity and let pi: V -> V parallel to G be the categorical quotient. Let N: = pi(-1)pi(0)(G) be the null cone of V, let H-0 be the subgroup of GL(V) which preserves the ideal T of N and let H be a Levi subgroup of H-0 containing G. We determine the identity component of H. In many cases we show that H = H-0. For adjoint representations we have H = H-0 and we determine H completely. We also investigate the subgroup G(F) of GL(V) preserving a fiber F of pi when V is an irreducible cofree G-module.