Cent U(n) and a construction of Lipsman-Wolf

TL;DR

This paper refines a representation theory approach to construct elements in the center of the universal enveloping algebra of a nilpotent Lie algebra within complex semisimple Lie groups, correcting a key lemma from prior work.

Contribution

It modifies the previous construction to successfully produce central elements in U(n), addressing an error in the original lemma from Lipsman-Wolf.

Findings

Corrected the construction of central elements in U(n)

Validated the modified approach through theoretical analysis

Enhanced understanding of the center in universal enveloping algebras

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 $\g$. The authors in [LW] introduce a very nice representation theory idea for the construction of certain elements in $\hbox{\rm cent}\,U(n)$. 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 the desired elements in $\hbox{\rm cent}\,U(\n)$.