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Twisted Whittaker modules over \({U_q}({\mathfrak{g}\mathfrak{l}_{n + 1}})\)

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Abstract

As is well known, the classical nonsingular Whittaker modules over quantum groups cannot be defined for non A1-type. In this paper, by choosing different generators for the quantum group \({U_q}({\mathfrak{g}\mathfrak{l}_{n + 1}})\), we introduce and study the twisted Whittaker modules over \({U_q}({\mathfrak{g}\mathfrak{l}_{n + 1}})\). We classify all the simple twisted Whittaker modules with nonsingular Whittaker functions. This agrees Kostant’s results on Whittaker modules for the simple complex Lie algebras \({\mathfrak{s}\mathfrak{l}_{n + 1}}\) as q approaches 1.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11871249, 11871190 and 12171155) and Natural Sciences and Engineering Research Council of Canada (Grant No. 311907-2015). The authors thank the referees for their nice suggestions.

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Authors and Affiliations

  1. Institute of Applied System Analysis, Jiangsu University, Zhenjiang, 212013, China

    Limeng Xia

  2. Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, N2L 3C5, Canada

    Kaiming Zhao

  3. School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, 050024, China

    Kaiming Zhao

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  1. Limeng Xia
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  2. Kaiming Zhao
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Correspondence to Kaiming Zhao.

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Xia, L., Zhao, K. Twisted Whittaker modules over \({U_q}({\mathfrak{g}\mathfrak{l}_{n + 1}})\). Sci. China Math. 66, 2191–2202 (2023). https://doi.org/10.1007/s11425-022-2079-3

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  • DOI: https://doi.org/10.1007/s11425-022-2079-3

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