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

Abstract: Let $G$ be a semisimple Lie group and let $\g =\n_- +\hh +\n$ be a triangular decomposition of $\g= \hbox{Lie}\,G$. Let $\b =\hh +\n$ and let $H,N,B$ be Lie subgroups of $G$ corresponding respectively to $\hh,\n$ and $\b$. We may identify $\n_-$ with the dual space to $\n$. The coadjoint action of $N$ on $\n_-$ extends to an action of $B$ on $\n_-$. There exists a unique nonempty Zariski open orbit $X$ of $B$ on $\n_-$. Any $N$-orbit in $X$ is a maximal coadjoint orbit of $N$ in $\n_-$. 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 $S(\n)^{\n}$ as a polynomial ring generated by the prime polynomials of $H$-weight vectors in $S(\n)^{\n}$. It also leads tothe multiplicity 1 of $H$ weights in $S(\n)^{\n}$.