Introduction to the relative Langlands program

Abstract

The aim of these notes is to give an overview of several aspects of what has come to be called the relative Langlands program, a theme that takes its origin in the study of automorphic periods and their relations to particular cases of Langlands functoriality and special values of (automorphic) $L$-functions. Following the work of Sakellaridis and Sakellaridis-Venkatesh, we emphasize the unifying role played by spherical varieties and harmonic analysis on them. We also review various instances of the phenomena, discovered by Jacquet, Gan-Gross-Prasad, Ichino-Ikeda and many others, that motivated these developments.