PBW theory for quantum affine algebras

Let U_q́.g/ be a quantum affine algebra of arbitrary type and let C_g⁰ be Hernandez-Leclerc’s category. We can associate the quantum affine Schur–Weyl duality functor F_D to a duality datum D in C_g⁰. In this paper, 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 ƒ, ƒ^z and ƒ¹ introduced by the authors. We next define the reflections S_k and S_k^-1 acting on strong duality data D. We prove that if D is a strong (resp. complete) duality datum, then S_k.D / and S_k^-1.D / are also strong (resp. complete) duality data. This allows us to make new strong (resp. complete) duality data by applying the reflections S_k and S_k^-1 from known strong (resp. complete) duality data. We finally introduce the notion of affine cuspidal modules in C_g⁰ by using the duality functor F_D, and develop the cuspidal module theory for quantum affine algebras similar to the quiver Hecke algebra case. When D is complete, we show that all simple modules in C_g⁰ can be constructed as the heads of ordered tensor products of affine cuspidal modules. We further prove that the ordered tensor products of affine cuspidal modules have the unitriangularity property. This generalizes the classical simple module construction using ordered tensor products of fundamental modules.