On the algebraic set of singular elements in a complex simple Lie algebra

Let $G$ be a complex simple Lie group and let $\g = \hbox{\rm Lie}\,G$. Let $S(\g)$ be the $G$-module of polynomial functions on $\g$ and let $\hbox{\rm Sing}\,\g$ be the closed algebraic cone of singular elements in $\g$. Let ${\cal L}\s S(\g)$ be the (graded) ideal defining $\hbox{\rm Sing}\,\g$ and let $2r$ be the dimension of a $G$-orbit of a regular element in $\g$. Then ${\cal L}^k = 0$ for any $k<r$. On the other hand, there exists a remarkable $G$-module $M\s {\cal L}^r$ which already defines $\hbox{\rm Sing}\,\g$. The main results of this paper are a determination of the structure of $M$.