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We give new criteria for the irreducibility of parabolic induction on the general linear group and its inner forms over a local non-archimedean field. In particular, we give a necessary and sufficient condition when the inducing data is of the form π ⊗ σ where π is a ladder representation and σ is an arbitrary irreducible representation. As an application we simplify the proof of the classification of the unitary dual. CONTENTS

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

We study a special class of irreducible representations of GL n over a local non-Archimedean field which we call ladder representations. This is a natural class in the admissible dual which contains the Speh representations. We show that the Tadić determinantal formula is valid for this class and analyze the standard modules pertaining to these representations.

Abstract. We formulate an analogue of the Ichino-Ikeda conjectures for the WhittakerFourier coefficients of automorphic forms on quasi-split reductive groups. This sharpens the conjectures of Sakellaridis-Venkatesh in the case at hand.