The Gelfand-Zeitlin integrable system and K-orbits on the flag variety

In this expository paper, we provide an overview of the Gelfand-Zeiltin integrable system on the Lie algebra of $n\times n$ complex matrices $\text{fgl}(n,\text{C})$ introduced by Kostant and Wallach in 2006. We discuss results concerning the geometry of the set of strongly regular elements, which consists of the points where Gelfand-Zeitlin flow is Lagrangian. We use the theory of $K_n=GL(n-1,\text{C})\times GL(1,\text{C})$-orbits on the flag variety ${\mathcal{B}}_n$ of $GL(n,\text{C})$ to describe the strongly regular elements in the nilfiber of the moment map of the system. We give an overview of the general theory of orbits of a symmetric subgroup of a reductive algebraic group acting on its flag variety, and illustrate how the general theory can be applied to understand the specific example of $K_n$ and $GL(n,\text{C})$.