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We prove that the Robinson-Schensted-Knuth correspondence is a$\gl_{\infty}$-crystal isomorphism between two realizations of the crystalgraph of a generalized Verma module with respect to a maximal parabolicsubalgebra of $\gl_{\infty}$. %This extends the previously known result that%the RSK correspondence is an isomorphism of bicrystals or double %crystals. Aflagged version of the RSK correspondence is derived in a natural way bycomputing a Demazure crystal graph of a generalized Verma module. As anapplication, we discuss a relation between a Demazure crystal and planepartitions with a bounded condition.

Demazure crystals of generalized Verma modules and a flagged RSK correspondence