PBW theoretic approach to the module category of quantum affine algebras

Let U-q'(g) be a quantum affine algebra of untwisted affine ADE type and let (0)(Cg) be Hernandez-Leclerc's category. For a duality datum D in (0)(Cg), we denote by F-D the quantum affine Weyl-Schur duality functor. We give a sufficient condition for a duality datum D to provide the functor F-D sending simple modules to simple modules. Moreover, under the same condition, the functor F-D has compatibility with the new invariants introduced by the authors. Then we introduce the notion of cuspidal modules in (0)(Cg), and show that all simple modules in( Cg)(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.