  We extend Schur-Weyl duality to an arbitrary level $l \geq 1$, the case $l=1$
recovering the classical duality between the symmetric and general linear
groups. In general, the symmetric group is replaced by the degenerate
cyclotomic Hecke algebra over $\C$ parametrized by a dominant weight of level
$l$ for the root system of type $A_\infty$. As an application, we prove that
the degenerate analogue of the quasi-hereditary cover of the cyclotomic Hecke
algebra constructed by Dipper, James and Mathas is Morita equivalent to certain
blocks of parabolic category $\mathcal{O}$ for the general linear Lie algebra.
