PBW theoretic approach to the module category of quantum affine algebras

Let $U_q(mathfrak{g})$ be a quantum affine algebra of untwisted affine ADE type and let $mathcal{C}^0_{mathfrak{g}}$ be Hernandez-Leclercs category. For a duality datum $mathcal{D}$ in $mathcal{C}^0_{mathfrak{g}}$, 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 $mathcal{C}^0_{mathfrak{g}}$, and show that all simple modules in $mathcal{C}^0_{mathfrak{g}}$ can be constructed as the heads of ordered tensor products of cuspidal modules.