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 $\mathcal{C}_{\mathfrak{g}}^{0}$ 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 a sufficient condition for a duality datum $\mathcal{D}$ to provide the functor $\mathcal{F}_{\mathcal{D}}$ sending simple modules to simple modules. Moreover, under the same condition, the functor $\mathcal{F}_{\mathcal{D}}$ has compatibility with the new invariants introduced by the authors. 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. We next state that the ordered tensor products of cuspidal modules have the unitriangularity property.