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.