Let E/F be a quadratic extension of number fields. We study periods and regularized periods of cusp forms and Eisenstein series on GL n (A E ) 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. CONTENTS

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Abstract. We show that every irreducible representation in the discrete automorphic spectrum of GL n (A) admits a non vanishing mixed (Whittaker-symplectic) period integral. The analog local problem is a study of models first considered by Klyachko over a finite field. Locally, we show that for a p-adic field F every irreducible, unitary representation of GL n (F ) has a Klyachko model.

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