Lagrangian fibers of Gelfand-Cetlin systems

A Gelfand-Cetlin system is a completely integrable system defined on a partial flag manifold whose image is a rational convex polytope called a Gelfand-Cetlin polytope. Motivated by the study of Nishinou-Nohara-Ueda on the Floer theory of Gelfand-Cetlin systems, we provide a detailed description of topology of Gelfand-Cetlin fibers. In particular, we prove that any fiber over an interior point of a k-dimensional face of the Gelfand-Cetlin polytope is an isotropic submanifold and is diffeomorphic to $(S^1)^k \times N$ for some smooth manifold $N$. We also prove that such $N$'s are exactly the vanishing cycles shrinking to points in the associated toric variety via the toric degeneration. We also devise an algorithm of reading off Lagrangian fibers from the combinatorics of the ladder diagram.