  Let H be a complex Lie group acting holomorphically on a complex analytic
space X such that the restriction to X_{\mathrm{red}} of every H-invariant
regular function on X 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.
