Vector fields and Luna strata

Let V be a G-module, where G is a complex reductive group. Let denote the categorical quotient. One can ask if the Luna stratification of Z is intrinsic. That is, if φ:Z→Z is any automorphism, does φ send strata to strata? In Kuttler and Reichstein (2008) [1], the answer was shown to be yes for V a direct sum of sufficiently many copies of a G-module W. We show that the answer is yes for almost all V. The key is to consider the vector fields on Z. Our methods also show that complex analytic automorphisms preserve the stratification.