** Abstract: **
Let
G -> \GL(V) be a finite-dimensional linear representation of
a reductive linear algebraic group G on a finite-dimensional
vector space V, defined over an algebraically closed field
of characteristic zero. The categorical quotient V//G
carries a natural stratification, due to D. Luna.
This paper addresses the following questions:
(i) Is the Luna stratification of X intrinsic? That is,
does every automorphism of V//G map each stratum
to another stratum?
(ii) Are the individual Luna strata in X intrinsic?
That is, does every automorphism of V//G maps each stratum
to itself?
In general, the Luna stratification is not intrinsic.
Nevertheless, we give positive answers to questions (i) and (ii)
for large classes of interesting representations.