The orbit method and analysis of automorphic forms

TL;DR

This paper develops a quantitative orbit method for automorphic forms, providing new asymptotic formulas for periods and special functions, with applications to automorphic representation theory and related conjectures.

Contribution

It introduces a quantitative orbit method aligned with microlocal analysis and applies it to derive asymptotic formulas for automorphic periods and special functions.

Findings

Asymptotic formula for averages of Gan--Gross--Prasad periods.

Asymptotic expansions for special functions from higher rank Lie group representations.

Application of Ratner's measure classification to automorphic form analysis.

Abstract

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.