PBW theoretic approach to the module category of quantum affine algebras

Let $U_q^{\mathrm{\prime }}(\mathfrak{g})$ be a quantum affine algebra of untwisted affine ADE type and let ${\mathcal{C}}_{\mathfrak{g}}^0$ be Hernandez-Leclerc’s category. For a duality datum $\mathcal{D}$ in ${\mathcal{C}}_{\mathfrak{g}}^0$, we denote by ${\mathcal{F}}_{\mathcal{D}}$ the quantum affine Weyl-Schur duality functor. We give sufficient conditions for a duality datum $\mathcal{D}$ to provide the functor ${\mathcal{F}}_{\mathcal{D}}$ sending simple modules to simple modules. Then we introduce the notion of cuspidal modules in ${\mathcal{C}}_{\mathfrak{g}}^0$, and show that all simple modules in ${\mathcal{C}}_{\mathfrak{g}}^0$ can be constructed as the heads of ordered tensor products of cuspidal modules.