The orbit method and analysis of automorphic forms

We develop the orbit method in a quantitative form, along lines of microlocal analysis, and apply it to analytic theory automorphic forms. Our main global application is an asymptotic formula for averages Gan--Gross--Prasad periods arbitrary rank. The form on larger group held fixed, while that smaller varies over family size roughly fourth root conductors corresponding $L$-functions. Ratner's results measure classification provide important input proof. local include expansions certain special functions arising from representations higher rank Lie groups, such as relative characters defined by matrix coefficient integrals Ichino--Ikeda conjecture.