Center $U(\mathfrak{n})$, Cascade of Orthogonal Roots, and a Construction of Lipsman–Wolf

Abstract

Let G be a complex simply-connected semisimple Lie group and let $\mathfrak{g} =\mathrm{ Lie}\,G$. Let $\mathfrak{g} = \mathfrak{n}_{-} + \mathfrak{h} + \mathfrak{n}$ be a triangular decomposition of $\mathfrak{g}$. One readily has that $\mathrm{Cent}\,U(\mathfrak{n})$ is isomorphic to the ring $S{(\mathfrak{n})}^{\mathfrak{n}}$ of symmetric invariants. Using the cascade $\mathcal{B}$ of strongly orthogonal roots which we introduced some time ago, we then proved [K] that $S{(\mathfrak{n})}^{\mathfrak{n}}$ is a polynomial ring $\mathbb{C}[\xi _{1},\ldots,\xi _{m}]$, where m is the cardinality of $\mathcal{B}$. The authors in [LW] introduce a very nice representation-theoretic method for the construction of certain elements in $S{(\mathfrak{n})}^{\mathfrak{n}}$. 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 $S{(\mathfrak{n})}^{\mathfrak{n}}$ and use the [LW] result to prove a theorem of Tony Joseph.

Dedicated to Joe, a special friend and valued colleague