  We prove that the fiber ring of the space of invariant jet differentials of a
projective manifold is finitely generated on the regular locus. The question
has been partially worked out by Berczi-Kirwan in \cite{BK}, however our method
is different and complementary to them. The analytic automorphism group of
regular $k$-jets of analytic curves on a projective variety $X$ is a
non-reductive subgroup of the general linear group $GL_k \mathbb{C}$. In this
case the Chevalley theorem on the invariant polynomials in the fiber rings
fails in general. Thus the analysis of Cartan subalgebras of the Lie algebra
and its Weyl group requires extra methods. We employ some methods of
non-reductive Lie theory together with basic results obtained in \cite{BK} to
prove the finite generation of the stalk ring at a regular point.
