On top Fourier coefficients of certain automorphic representations of \({\mathrm {GL}}_n\)

Abstract

We study the top Fourier coefficients of isobaric automorphic representations of \({\mathrm {GL}}_n({\mathbb {A}})\) of the form

$$\begin{aligned} \Pi _{\underline{s}} = {\mathrm {Ind}}^{{\mathrm {GL}}_n({\mathbb {A}})}_{P({\mathbb {A}})} \Delta (\tau _1,b_1) |\cdot |^{s_1} \otimes \cdots \otimes \Delta (\tau _r,b_r) |\cdot |^{s_r}, \end{aligned}$$

where \(s_i\in {\mathbb {C}}\), \(\Delta (\tau _i,b_i)\)’s are Speh representations in the discrete spectrum of \({\mathrm {GL}}_{a_ib_i}({\mathbb {A}})\) with \(\tau _i\)’s being unitary cuspidal representations of \({\mathrm {GL}}_{a_i}({\mathbb {A}})\), and \(n = \sum _{i=1}^r a_ib_i\). In particular, we prove a part of a conjecture of Ginzburg, and also a conjecture of Jiang under certain assumptions. The result of this paper will facilitate the study of automorphic forms of classical groups occurring in the discrete spectrum.

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Acknowledgements

We would like to thank Dihua Jiang for his interest in this work and for the valuable suggestions and constant encouragement. We also thank Yuanqing Cai for helpful communication on the result in his paper [3]. Finally, we are grateful to the referee for pointing out a mistake in the main result of a previous verion, the careful reading of our manuscript, and the very useful comments and suggestions, which improve the exposition of the paper much.

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Correspondence to Bin Xu.

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The research of the first named author is partially supported by NSF Grants DMS-1702218, DMS-1848058, and by start-up funds from the Department of Mathematics at Purdue University. The second named author is partially supported by NSFC Grant No.11501382 and by the Fundamental Research Funds for the Central Universities.

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Liu, B., Xu, B. On top Fourier coefficients of certain automorphic representations of \({\mathrm {GL}}_n\). manuscripta math. 164, 1–22 (2021). https://doi.org/10.1007/s00229-019-01176-z

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Mathematics Subject Classification

  • Primary 11F30
  • 22E55
  • Secondary 22E50
  • 11F70