The orbit method and analysis of automorphic forms

Paul D. Nelson Departement Mathematik, Eidgenössische Technische Hochschule, Zürich, Switzerland Akshay Venkatesh Institute for Advanced Study, Princeton, New Jersey, U.S.A.

TBD mathscidoc:2203.43006

Acta Mathematica, 226, (1), 1-209, 2021.3
We develop the orbit method in a quantitative form, along the lines of microlocal analysis, and apply it to the analytic theory of automorphic forms. Our main global application is an asymptotic formula for averages of Gan-Gross-Prasad periods in arbitrary rank. The automorphic form on the larger group is held fixed, while that on the smaller group varies over a family of size roughly the fourth root of the conductors of the corresponding L-functions. Ratner’s results on measure classification provide an important input to the proof. Our local results include asymptotic expansions for certain special functions arising from representations of higher-rank Lie groups, such as the relative characters defined by matrix coefficient integrals as in the Ichino-Ikeda conjecture.
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@inproceedings{paul2021the,
  title={The orbit method and analysis of automorphic forms},
  author={Paul D. Nelson, and Akshay Venkatesh},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220310110633064045921},
  booktitle={Acta Mathematica},
  volume={226},
  number={1},
  pages={1-209},
  year={2021},
}
Paul D. Nelson, and Akshay Venkatesh. The orbit method and analysis of automorphic forms. 2021. Vol. 226. In Acta Mathematica. pp.1-209. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220310110633064045921.
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