  We prove that the Robinson-Schensted-Knuth correspondence is a
$\gl_{\infty}$-crystal isomorphism between two realizations of the crystal
graph of a generalized Verma module with respect to a maximal parabolic
subalgebra of $\gl_{\infty}$. %This extends the previously known result that
%the RSK correspondence is an isomorphism of bicrystals or double %crystals. 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.
