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