On representations distinguished by unitary groups

Abstract

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

On a new functional equation for local integrals

“…Note that the characterization of generic irreducible representations of GL n (C) which are distinguished by a unitary group was given in [FLO12]. It is likely that the methods of [loc.…”

A nonabelian trace formula

“…Provided that N is sufficiently large this is bounded by a constant depending on N times The proposition now follows from [6] (Theorem 10.2).…”

“…The idea is to use the special Hecke operators W g to build a weight consisting of the period of a cusp form over a unitary group into the trace formula. This weight isolates the representations of G(A F ) invariant under ι by work of Jacquet (we actually require the refinements of [6]). These V are then used in § 5.2 to prove our main result, Theorem 1.2.…”

Local unitary periods and relative discrete series

“…As above, let G = GL 2n (E) and H = U E/F,w (F ), where n ≥ 2. Building on the work of Jacquet [16] and Feigon-Lapid-Offen [8], Beuzart-Plessis has shown that the Plancherel formula for H\G is the push-forward of the Whittaker-Plancherel formula for GL 2n (F ) via quadratic base change. As a consequence, the discrete spectrum of H\G consists of the quadratic base changes of the discrete series of GL 2n (F ).…”

“…In this case, the representation π is the automorphic induction of the discrete series τ , see[8, Section 3.2].…”