On representations distinguished by unitary groups

Let E/F be a quadratic extension of number fields. We study periods and regularized periods of cusp forms and Eisenstein series on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\operatorname {GL}_{n}( \mathbf {A}_{E})$\end{document} over a unitary group of a Hermitian form with respect to E/F. We provide factorization for these periods into locally defined functionals, express these factors in terms of suitably defined local periods and characterize global distinction. We also study in detail the analogous local question and analyze the space of invariant linear forms under a unitary group.