Title: Lagrangian fibers of Gelfand-Cetlin systems,
Abstract: Motivated by the study of Nishinou-Nohara-Ueda on the Floer thoery of
Gelfand-Cetlin systems over complex partial flag manifolds, we provide a
complete description of the topology of Gelfand-Cetlin fibers. We prove that
all fibers are \emph{smooth} isotropic submanifolds and give a complete
description of the fiber to be Lagrangian in terms of combinatorics of
Gelfand-Cetlin polytope. Then we study (non-)displaceability of Lagrangian
fibers. After a few combinatorial and numercal tests for the displaceability,
using the bulk-deformation of Floer cohomology by Schubert cycles, we prove
that every full flag manifold $\mathcal{F}(n)$ ($n \geq 3$) with a monotone
Kirillov-Kostant-Souriau symplectic form carries a continuum of
non-displaceable Lagrangian tori which degenerates to a non-torus fiber in the
Hausdorff limit. In particular, the Lagrangian $S^3$-fiber in $\mathcal{F}(3)$
is non-displaceable the question of which was raised by Nohara-Ueda who
computed its Floer cohomology to be vanishing.