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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 decompositionof $\g$. One readily has that $\hbox{\rm Cent}\,U(\n)$ is isomorphic to thering $S(\n)^{\n}$ of symmetric invariants. Using the cascade ${\cal B}$ ofstrongly 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-theoreticmethod 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 modifythe construction so as to yield these elements in $S(\n)^{\n}$ and use the [LW]result to prove a theorem of Tony Joseph.

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