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

Let $G$ be a complex simple Lie group and let $\text{g}=\text{Lie}G$. Let $S(\text{g})$ be the $G$-module of polynomial functions on $\text{g}$ and let $\hbox{\rm Sing},\g$betheclosedalgebraicconeofsingularelementsin$\g$\mathrm{.}Let${\cal L}\s S(\g)$bethe(graded)idealdefining$\hbox{\rm Sing},\g$andlet$2r$bethedimensionofa$G$-orbitofaregularelementin$\g$\mathrm{.}Then${\cal L}^k = 0$forany$k<r$\mathrm{.}Ontheotherhand,thereexistsaremarkable$G$-module$M\s {\cal L}^r$whichalreadydefines$\hbox{\rm Sing},\g$. The main results of this paper are a determination of the structure of $M$.