Regularized periods and the global Gan–Gross–Prasad conjecture: the case of U(n+2r)×U(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U(n+2r) \times U(n)$$\end{document}

In this paper, we introduce regularized trilinear periods on certain non-reductive groups. It has two direct applications. Firstly, it enables us to define the regularized Bessel periods and the regularized Fourier–Jacobi periods for all classical and metaplectic groups. Secondly, by using the properties of the regularized Fourier–Jacobi periods, we can prove one direction of the global Gan–Gross–Prasad conjecture on skew-Hermitian unitary groups for tempered cases.