PBW theory for quantum affine algebras

Let $U_q^{\mathrm{\prime }}(\mathfrak{g})$ be a quantum affine algebra of arbitrary type and let ${\mathcal{C}}_{\mathfrak{g}}$ be Hernandez-Leclerc’s category. We can associate the quantum affine Schur-Weyl duality functor $F_D$ to a duality datum $D$ in ${\mathcal{C}}_{\mathfrak{g}}$. 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 $\mathrm{\Lambda }$ and ${\mathrm{\Lambda }}^{\mathrm{\infty }}$ introduced by the authors. We next define the reflections ${\mathcal{S}}_k$ and ${\mathcal{S}}_k^{-1}$ acting on strong duality data $D$. We prove that if $D$ is a strong (resp.\ complete) duality datum, then ${\mathcal{S}}_k(D)$ and ${\mathcal{S}}_k^{-1}(D)$ are also strong (resp.\ complete ) duality data. We finally introduce the notion of affine cuspidal modules in ${\mathcal{C}}_{\mathfrak{g}}$ by using the duality functor $F_D$, and develop the cuspidal module theory for quantum affine algebras similarly to the quiver Hecke algebra case.