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

TL;DR

This paper characterizes the orbit structure of a Lie group action on matrices derived from Gelfand-Zeitlin theory, providing a complete description of all maximal orbits and their role in the geometry of regular adjoint orbits.

Contribution

It extends previous work by fully describing all orbits of the Gelfand-Zeitlin group on matrices, not just a subset, revealing the complete polarization structure.

Findings

All orbits of dimension n choose 2 are classified.

The orbit structure relates to the polarization of regular adjoint orbits.

Provides a comprehensive description of the Gelfand-Zeitlin group action.

Abstract

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