  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 $S(\n)^{\n}$.
