PBW theory for quantum affine algebras

Abstract

Let $U_q^{\prime }(\mathfrak{g})$ be a quantum affine algebra of arbitrary type and let ${\mathcal{C}}_{\mathfrak{g}}^0$ be Hernandez-Leclerc’s category. We can associate the quantum affine Schur–Weyl duality functor ${\mathcal{F}}_{\mathcal{D}}$ to a duality datum $\mathcal{D}$ in ${\mathcal{C}}_{\mathfrak{g}}^0$. In this paper, we introduce the notion of a strong (complete) duality datum $\mathcal{D}$ and prove that, when $\mathcal{D}$ is strong, the induced duality functor ${\mathcal{F}}_{\mathcal{D}}$ sends simple modules to simple modules and preserves the invariants $\Lambda$, $\tilde{\Lambda }$ and ${\Lambda }^{\infty }$ introduced by the authors. We next define the reflections ${\mathcal{S}}_k$ and ${\mathcal{S}}_k^{-1}$ acting on strong duality data $\mathcal{D}$. We prove that if $\mathcal{D}$ is a strong (resp.\ complete) duality datum, then ${\mathcal{S}}_k(\mathcal{D})$ and ${\mathcal{S}}_k^{-1}(\mathcal{D})$ are also strong (resp. complete) duality data. This allows us to make new strong (resp. complete) duality data by applying the reflections ${\mathcal{S}}_k$ and ${\mathcal{S}}_k^{-1}$ from known strong (resp. complete) duality data. We finally introduce the notion of affine cuspidal modules in ${\mathcal{C}}_{\mathfrak{g}}^0$ by using the duality functor ${\mathcal{F}}_{\mathcal{D}}$, and develop the cuspidal module theory for quantum affine algebras similar to the quiver Hecke algebra case. When $\mathcal{D}$ is complete, we show that all simple modules in ${\mathcal{C}}_{\mathfrak{g}}^0$ 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.