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Cuspidal components of Siegel modular forms for large discrete series representations of \(\textrm{Sp}_4({\mathbb {R}})\)

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Abstract

In this paper, we consider automorphic forms on \(\textrm{Sp}_4({\mathbb {A}}_{\mathbb {Q}})\) which generate large discrete series representations of \(\textrm{Sp}_4({\mathbb {R}})\) as \((\mathfrak {sp}_4({\mathbb {R}}),K_\infty )\)-modules. We determine the cuspidal components and the structure of the space of such automorphic forms.

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Notes

  1. As well as this paper deals with a general case it includes quite complicated calculations and would therefore take a long time toward publication.

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Acknowledgements

This work was supported by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University. The first author is partially supported by AIP Challenge of JST CREST JPMJCR14D6, JST CREST JPMJCR14D6 and CREST JPMJCR2113, Japan. The second named author was partially supported by Grand-in-Aid for Scientific Research (C) 19K03431, Japan Society for the Promotion of Science. We are very grateful to Professor Taku Ishii for his generosity to use a part of the results in the paper about the Whittaker functions on \(\textrm{Sp}_4(\mathbb {R})\) attached to degenerate characters, being prepared jointly with the second named author. We would like to express our profound gratitude to the referee for careful read of our paper and fruitful suggestions to enhance the paper.

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Authors and Affiliations

  1. NTT Institute for Fundamental Mathematics, NTT Communication Science Laboratories, NTT Corporation, 2-4, Hikaridai, Seika-cho, Soraku-gun, Kyoto, 619-0237, Tokyo, Japan

    Shuji Horinaga

  2. Department of Mathematics, Faculty of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo, 169-8555, Japan

    Hiro-aki Narita

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  1. Shuji Horinaga
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  2. Hiro-aki Narita
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Correspondence to Shuji Horinaga.

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Horinaga, S., Narita, Ha. Cuspidal components of Siegel modular forms for large discrete series representations of \(\textrm{Sp}_4({\mathbb {R}})\). manuscripta math. 174, 159–202 (2024). https://doi.org/10.1007/s00229-023-01513-3

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