  Let H be a complex Lie group acting holomorphically on a complex analytic
space X such that every H-invariant regular function on X_{red} is constant. We
prove that an H-equivariant holomorphic vector bundle E over X is H-finite,
meaning f_1(E)= f_2(E) as H-equivariant bundles for two distinct polynomials
f_1 and f_2 whose coefficients are nonnegative integers, if and only if the
pullback of E along some H-equivariant finite \'etale covering of X is trivial
as an H-equivariant bundle.
