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

Abstract

In recent work (\cite{KW1},\cite{KW2}), Kostant and Wallach construct an action of a simply connected Lie group $A\simeq {\mathbb{C}}^{\left(\genfrac{}{}{0ex}{}{n}{2}\right)}$ 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 $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ 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 $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ and thus all polarizations of regular adjoint orbits obtained using Gelfand-Zeitlin theory.

30 pages: Version 2 contains a stronger result in section 5.3 (Theorem 5.15)

References in corpus (1)

• The Gelfand-Zeitlin integrable system and its action on generic elements of gl(n) and so(n)

Cited by in corpus (3)

• The Gelfand-Zeitlin integrable system and its action on generic elements of gl(n) and so(n)
• On Algebraic Integrability of Gelfand-Zeitlin fields
• K-orbits on the flag variety and strongly regular nilpotent matrices