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

TL;DR

This paper studies the coadjoint structure of the nilradical of a Borel subgroup in a semisimple Lie group, introducing a cascade of orthogonal roots to describe orbit decompositions and polynomial invariants.

Contribution

It introduces a cascade of orthogonal roots to explicitly describe the coadjoint orbit structure and polynomial invariants of the nilradical of a Borel subgroup.

Findings

Decomposition of an open orbit into a product involving a cross-section.

Structure of the symmetric invariants as a polynomial ring.

Proof of multiplicity-one property of H-weights in symmetric invariants.

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