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 a link between the existence of open coadjoint orbits and the Weyl group structure.

Contribution

It proves that the maximal coadjoint orbit of a Borel subgroup has codimension related to the rank and root system, connecting algebraic invariants with geometric orbit properties.

Findings

Maximal coadjoint orbit of B has codimension  - m.

Open coadjoint orbit exists iff  = 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 open coadjoint orbit are due to Piatetski--Shapiro in his counterexample construction of a bounded complex homogeneous domain which is not of Cartan type.