On Jacobian Ideals Invariant by a Reducible $s\ell (2, \mathbf {C})$ Action

This paper deals with a reducible $s\ell (2, \mathbf {C})$ action on the formal power series ring. The purpose of this paper is to confirm a special case of the Yau Conjecture: suppose that $s\ell (2, \mathbf {C})$ acts on the formal power series ring via $(0.1)$. Then $I(f)=(\ell _{i_{1}})\oplus (\ell _{i_{2}})\oplus \cdots \oplus (\ell _{i_{s}})$ modulo some one dimensional $s\ell (2, \mathbf {C})$ representations where $(\ell _{i})$ is an irreducible $s\ell (2, \mathbf {C})$ representation of dimension $\ell _{i}$ or empty set and $\{\ell _{i_{1}},\ell _{i_{2}},\ldots ,\ell _{i_{s}}\}\subseteq \{\ell _{1},\ell _{2},\ldots ,\ell _{r}\}$. Unlike classical invariant theory which deals only with irreducible action and 1--dimensional representations, we treat the reducible action and higher dimensional representations succesively.