In this talk, we will introduce a new technique of isolating the cuspidal part, or more general cuspidal components, from the L^2 spectrum on the spectral side of the trace formula. As an application, we prove the global Gan-Gross-Prasad and Ichino-Ikeda conjectures for unitary groups in all stable cases.

This is a joint work with Raphaël Beuzart-Plessis, Wei Zhang, and Xinwen Zhu.