The orbit structure of the Gelfand-Zeitlin group on n x n matrices

In recent work (\cite{KW1},\cite{KW2}), Kostant and Wallach construct an action of a simply connected Lie group $A\simeq {\mathbb{C}}^{\left(\genfrac{}{}{0px}{}{n}{2}\right)}$ on $gl(n)$ using a completely integrable system derived from the Poisson analogue of the Gelfand-Zeitlin subalgebra of the enveloping algebra. In \cite{KW1}, the authors show that $A$-orbits of dimension $\left(\genfrac{}{}{0px}{}{n}{2}\right)$ form Lagrangian submanifolds of regular adjoint orbits in $gl(n)$. They describe the orbit structure of $A$ on a certain Zariski open subset of regular semisimple elements. In this paper, we describe all $A$-orbits of dimension $\left(\genfrac{}{}{0px}{}{n}{2}\right)$ and thus all polarizations of regular adjoint orbits obtained using Gelfand-Zeitlin theory.