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Large values of cusp forms on \(\mathrm {GL}_n\)

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Abstract

We establish the transition behavior of Jacquet–Whittaker functions on split semi-simple Lie groups. As a consequence, we show that for certain finite volume Riemannian manifolds, the local bound for normalized Laplace eigenfunctions does not hold globally.

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Acknowledgements

We would like to thank Michael Berry, Erez Lapid, Elon Lindenstrauss, Simon Marshall, Philippe Michel, and Andre Reznikov for helpful discussions. We thank the referees for many constructive comments that improved the quality of the paper. Some of the results of this paper were first announced at the Oberwolfach workshop 1135 on the analytic theory of automorphic forms and further presented at various other meetings, e.g. the Banff workshop on Whittaker functions and Physics, and the 17th Midrasha Mathematicae at Jerusalem in honor of Peter Sarnak. We thank the organizers for these invitations and the participants for their helpful comments.

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  1. Institut Galilée, Universite Paris 13, 93430, Villetaneuse, France

    Farrell Brumley

  2. Department of Mathematics, Cornell University, Malott Hall, Ithaca, NY, 14853-4201, USA

    Nicolas Templier

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Brumley, F., Templier, N. Large values of cusp forms on \(\mathrm {GL}_n\). Sel. Math. New Ser. 26, 63 (2020). https://doi.org/10.1007/s00029-020-00589-z

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