The cascade of orthogonal roots and the coadjoint structure of the nilradical of a Borel subgroup of a semisimple Lie group

Let $G$ be a semisimple Lie group and let $\text{g}={\text{n}}_{-}+\text{hh}+\text{n}$ be a triangular decomposition of $\text{g}=\text{Lie}G$. Let $\text{b}=\text{hh}+\text{n}$ and let $H,N,B$ be Lie subgroups of $G$ corresponding respectively to $\text{hh},\text{n}$ and $\text{b}$. We may identify ${\text{n}}_{-}$ with the dual space to $\text{n}$. The coadjoint action of $N$ on ${\text{n}}_{-}$ extends to an action of $B$ on ${\text{n}}_{-}$. There exists a unique nonempty Zariski open orbit $X$ of $B$ on ${\text{n}}_{-}$. Any $N$-orbit in $X$ is a maximal coadjoint orbit of $N$ in ${\text{n}}_{-}$. The cascade of orthogonal roots defines a cross-section \r_-^{\times} of the set of such orbits leading to a decomposition X = N/R\times \r_-^{\times}. This decomposition, among other things, establishes the structure of $S(\text{n}{)}^{\text{n}}$ as a polynomial ring generated by the prime polynomials of $H$-weight vectors in $S(\text{n}{)}^{\text{n}}$. It also leads tothe multiplicity 1 of $H$ weights in $S(\text{n}{)}^{\text{n}}$.