  In this paper we construct a family of exact functors from the category of
Whittaker modules of the simple complex Lie algebra of type $A_n$ to the
category of finite-dimensional modules of the graded affine Hecke algebra of
type $A_\ell$. Using results of Backelin and of Arakawa-Suzuki, we prove that
these functors map standard modules to standard modules (or zero) and simple
modules to simple modules (or zero). Moreover, we show that each simple module
of the graded affine Hecke algebra appears as the image of a simple Whittaker
module. Since the Whittaker category contains the BGG category $\mathcal{O}$ as
a full subcategory, our results generalize results of Arakawa-Suzuki, which in
turn generalize Schur-Weyl duality between finite-dimensional representations
of $SL_n(\mathbb{C})$ and representations of the symmetric group $S_n$.
