## Abstract

Let U_{q́}.g/ be a quantum affine algebra of arbitrary type and let C_{g}^{0} be Hernandez-Leclerc’s category. We can associate the quantum affine Schur–Weyl duality functor F_{D} to a duality datum D in C_{g}^{0}. 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 ƒ^{1} 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_{g}^{0} 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_{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.

Original language | English |
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Pages (from-to) | 2679-2743 |

Number of pages | 65 |

Journal | Journal of the European Mathematical Society |

Volume | 26 |

Issue number | 7 |

DOIs | |

State | Published - 2024 |

## Keywords

- Affine cuspidal modules
- Hernandez–Leclerc category
- PBW theory
- quantum affine algebra
- quantum affine Schur–Weyl duality
- quiver Hecke algebra