  Let $U_q'(\mathfrak{g})$ be a quantum affine algebra of arbitrary type and
let $\mathcal{C}_{\mathfrak{g}}$ be Hernandez-Leclerc's category. We can
associate the quantum affine Schur-Weyl duality functor $F_D$ to a duality
datum $D$ in $\mathcal{C}_{\mathfrak{g}}$. We introduce the notion of a strong
(complete) duality datum $D$ and prove that, when $D$ is strong, the induced
duality functor $F_D$ sends simple modules to simple modules and preserves the
invariants $\Lambda$ and $\Lambda^\infty$ introduced by the authors. We next
define the reflections $\mathcal{S}_k$ and $\mathcal{S}^{-1}_k$ acting on
strong duality data $D$. We prove that if $D$ is a strong (resp.\ complete)
duality datum, then $\mathcal{S}_k(D)$ and $\mathcal{S}_k^{-1}(D)$ are also
strong (resp.\ complete ) duality data. We finally introduce the notion of
affine cuspidal modules in $\mathcal{C}_{\mathfrak{g}}$ by using the duality
functor $F_D$, and develop the cuspidal module theory for quantum affine
algebras similarly to the quiver Hecke algebra case.

Representation Theory;Quantum Algebra