Acta Mathematica

Volume 226 (2021)

Number 1

The orbit method and analysis of automorphic forms

Pages: 1 – 209



Paul D. Nelson (Departement Mathematik, Eidgenössische Technische Hochschule, Zürich, Switzerland)

Akshay Venkatesh (Institute for Advanced Study, Princeton, New Jersey, U.S.A.)


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.

Received 12 February 2019

Received revised 27 August 2020

Accepted 17 December 2020

Published 30 March 2021