Center of U(n), Cascade of Orthogonal Roots, and a Construction of Lipsman-Wolf

TL;DR

This paper explores the structure of symmetric invariants in the universal enveloping algebra of a Lie algebra, modifies a previous construction, and applies it to prove a theorem related to the Lipsman-Wolf construction.

Contribution

It provides a corrected construction of elements in symmetric invariants and applies this to prove a theorem of Tony Joseph.

Findings

Established that $S( )^{ }$ is a polynomial ring generated by elements associated with strongly orthogonal roots.

Modified the Lipsman-Wolf construction to produce elements in $S( )^{ }$ correctly.

Proved a theorem of Tony Joseph using the revised construction.

Abstract

Let $G$ be a complex simply-connected semisimple Lie group and let $\g=\hbox{\rm Lie}\,G$. Let $\g = \n_- +\hh + \n$ be a triangular decomposition of $\g$. One readily has that $\hbox{\rm Cent}\,U(\n)$ is isomorphic to the ring $S(\n)^{\n}$ of symmetric invariants. Using the cascade ${\cal B}$ of strongly orthogonal roots, some time ago we proved (see [K]) that $S(\n)^{\n}$ is a polynomial ring $\Bbb C[\xi_1,...,\xi_m]$ where $m$ is the cardinality of ${\cal B}$. The authors in [LW] introduce a very nice representation-theoretic method for the construction of certain elements in $S(\n)^{\n}$. A key lemma in [LW] is incorrect but the idea is in fact valid. In our paper here we modify the construction so as to yield these elements in $S(\n)^{\n}$ and use the [LW] result to prove a theorem of Tony Joseph.