On non-displaceable Lagrangian submanifolds in two-step flag varieties

We prove that the two-step flag variety $\mathcal{F}\mathrm{\ell }(1,n;n+1)$ carries a non-displaceable and non-monotone Lagrangian Gelfand–Zeitlin fiber diffeomorphic to $S^3\times T^{2n-4}$ and a continuum family of non-displaceable Lagrangian Gelfand–Zeitlin torus fibers when $n>2$.