Ideals of Coadjoint Orbits of Nilpotent Lie Algebras

For f a linear functional on a nilpotent Lie algebra g over a field of characteristic 0, let $J(f)$ be the ideal of all polynomials in $S(g)$ vanishing on the coadjoint orbit through f in ${g^\ast }$, and let $I(f)$ be the primitive ideal of Dixmier in the universal enveloping algebra $U(g)$, corresponding to the orbit. An inductive method is given for computing generators ${P_1}, \ldots ,{P_r}$ of $J(f)$ such that $\varphi {P_1}, \ldots ,\varphi {P_r}$ generate $I(f),\varphi$ being the symmetrization map from $S(g)$ to $U(g)$. Upper bounds are given for the number of variables in the polynomials ${P_i}$ and a counterexample is produced for upper bounds proposed by Kirillov.