The coadjoint structure of Borel subgroups and their nilradicals

TL;DR

This paper investigates the coadjoint orbits of Borel subgroups in complex semisimple Lie groups, establishing conditions under which these orbits are open and relating this to the Weyl group structure.

Contribution

It proves that the maximal coadjoint orbit of a Borel subgroup has a codimension linked to the rank and root system, connecting orbit structure to Weyl group properties.

Findings

Maximal coadjoint orbit of B has codimension e9 - m.

Open coadjoint orbit exists iff e9 = m, i.e., -1 in the Weyl group.

Nilpotent and semisimple groups cannot have open coadjoint orbits.

Abstract

Let $G$ be a complex simply-connected semisimple Lie group and let $\frak{g}= Lie G$. Let $\frak{g} = \frak{n}_- +\frak{h} + \frak{n}$ be a triangular decomposition of $\frak{g}$. One readily has that $Cent\,U({\frak n})$ is isomorphic to the ring $S({\frak n})^{{\frak\n}}$ of symmetric invariants. Using the cascade ${\cal B}$ of strongly orthogonal roots, some time ago we proved that $S({\frak n})^{{\frak n}$ is a polynomial ring $\Bbb C [\xi_1,...,\xi_m]$ where $m$ is the cardinality of ${\cal B}$. Using this result we establish that the maximal coadjoint of $N = exp \frak {n}$ has codimension $m$. Let $\frak {b}= \frak {h} + \frak {n}$ so that the corresponding subgroup $B$ is a Borel subgroup of $G$. Let $\ell = rank \frak{g}$. Then in this paper we prove the theorem that the maximal coadjoint orbit of $B$ has codimension $\ell - m$ so that the following statements (1) and (2) are equivalent: (1) -1 is in the Weyl group of $G$ (i.e., $\ell = m$), and (2), B has a nonempty open coadjoint orbit. We remark that a nilpotent or a semisimple group cannot have a nonempty open coadjoint orbit. Celebrated examples where a solvable Lie group has a nonempty coadjoint orbit are due to Piatetski--Shapiro in his counterexample construction of a bounded complex homogeneous domain which is not of Cartan type.