The Local Gan-Gross-Prasad Conjecture and Endoscopic Classification for Quasi-Split Unitary Groups

This monograph establishes a rigorous framework for the resolution of the local Gan-Gross-Prasad (GGP) conjecture for quasi-split unitary groups over non-Archimedean local fields. We leverage the endoscopic classification of representations of quasi-split unitary groups, as developed by Mok, to define the requisite local L-packets and Arthur packets. The core of the study focuses on the restriction of irreducible admissible representations from U(n) to U(n-1), analyzing the spectral decomposition of the associated Bessel periods. We provide a detailed exposition of the multiplicity-one theorem within Vogan packets, characterizing the unique constituent satisfying the local GGP condition in terms of local root numbers and epsilon factors of symplectic representations of the L-group. Furthermore, we discuss the implications of the stabilization of the trace formula and the geometric expansion of the relevant relative trace formulae, synthesizing the results of Beuzart-Plessis and Waldspurger to confirm the conjecture for tempered representations. The interplay between the local Langlands correspondence and the endoscopic character relations is elucidated to provide a complete classification of the spectrum.