## Abstract

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

Original language | English |
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Pages (from-to) | 33-37 |

Number of pages | 5 |

Journal | Proceedings of the Japan Academy Series A: Mathematical Sciences |

Volume | 97 |

Issue number | 6 |

DOIs | |

State | Published - 2021 |

### Bibliographical note

Funding Information:Acknowledgements. The research of Ka- shiwara was supported by Grant-in-Aid for Scientific Research (B) 15H03608, Japan Society for the Promotion of Science, the research of Kim was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIP) (NRF-2017R1C1B2007824) and the research of Oh was supported by the Ministry of Education of the Republic of Korea and the NRF of Korea (NRF-2019R1A2C4069647).

Publisher Copyright:

© The Japan Academy

## Keywords

- Cuspidal modules
- Hernandez-Leclerc category
- quantum affine Weyl-Schur duality
- quantum affine algebra
- quiver Hecke algebra