On Jacobian Ideals Invariant by a Reducible Action

This paper deals with a reducible action on the formal power series ring. The purpose of this paper is to confirm a special case of the Yau Conjecture: suppose that acts on the formal power series ring via . Then modulo some one dimensional representations where is an irreducible representation of dimension or empty set and . Unlike classical invariant theory which deals only with irreducible action and 1--dimensional representations, we treat the reducible action and higher dimensional representations succesively.