Let $G$G be a connected simple algebraic group over the complex numbers and $B$B a Borel subgroup. Let $\germ g$g be the Lie algebra of $G$G and $\germ b$b the Lie algebra of $B$B . A subspace of the nilradical of $\germ b$b which is stable under the action of $B$B is called a $B$B -stable ideal of the nilradical. It is called strictly positive if it intersects the simple root spaces trivially.
The author counts the number of strictly positive $B$B -stable ideals in the nilradical of a Borel subalgebra and proves that the set of minimal roots of any $B$B -stable ideal is conjugate by an element of the Weyl group to a subset of simple roots. He also counts the number of ideals whose minimal roots are conjugate to a fixed subset of simple roots.