The Gelfand–Zeitlin integrable system and K-orbits on the flag variety
In this paper, we provide an overview of the Gelfand–Zeitlin integrable system on the Lie algebra of n × n complex matrices \(\mathfrak{g}\mathfrak{l}(n, \mathbb{C})\) 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 the Gelfand–Zeitlin flow is Lagrangian. We use the theory of \(K_{n} = GL(n - 1,\mbox{ $\mathbb{C}$}) \times GL(1,\mbox{ $\mathbb{C}$})\)-orbits on the flag variety \(\mathcal{B}_{n}\) of \(GL(n,\mbox{ $\mathbb{C}$})\) 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).