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 $Ms {cal L}^r$ which already defines $hbox{rm Sing},g$. The main results of this paper are a determination of the structure of $M$.