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 $\fgl(n,\C)$ 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 $K_{n}=GL(n-1,\C)\times GL(1,\C)$-orbits on the flag variety $\mathcal{B}_{n}$ of $GL(n,\C)$ 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 $GL(n,\C)$.