Demazure crystals of generalized Verma modules and a flagged RSK correspondence

We prove that the Robinson–Schensted–Knuth correspondence is a gl∞-crystal isomorphism between two realizations of the crystal graph of a generalized Verma module with respect to a maximal parabolic subalgebra of gl∞. A flagged version of the RSK correspondence is derived in a natural way by computing a Demazure crystal graph of a generalized Verma module. As an application, we discuss a relation between a Demazure crystal and plane partitions with a bounded condition.