Unitary dual ofGL(n) at archimedean places and global Jacquet–Langlands correspondence

Abstract

In a paper by Badulescu [Global Jacquet-Langlands correspondence, multiplicity one and classification of automorphic representations, Invent. Math. 172 (2008), 383-438], results on the global Jacquet-Langlands correspondence, (weak and strong) multiplicity-one theorems and the classification of automorphic representations for inner forms of the general linear group over a number field were established, under the assumption that the local inner forms are split at archimedean places. In this paper, we extend the main local results of that article to archimedean places so that the above condition can be removed. Along the way, we collect several results about the unitary dual of general linear groups over R, C or H which are of independent interest.

Local root numbers, Bessel models, and a conjecture of Guo and Jacquet

“…In fact, for n = 1, these local epsilon factor conditions for π E are precisely the local root number conditions in (3). For general n, the condition that (1/2, π E,v ) = 1 for all v means that the member σ of the Vogan L-packet of σ which locally everywhere has an E-Bessel model is actually a representation of SO(2n+1), rather than just a pure inner form of SO(2n + 1), but for n > 1 the conditions (1/2, π E v ) = 1 do not suffice to guarantee σ is the generic representation σ of SO(2n + 1).…”

“…Let G and H be algebraic groups defined over F with common center Z, and suppose H is a closed subgroup of G. In this paper, a (cuspidal) automorphic representation means an irreducible unitary (cuspidal) automorphic representation. We say a cuspidal representation π of G(A) with trivial central character is H-distinguished if the period integral Let E/F be a quadratic extension of number fields and X(E : F ) denote the set of isomorphism classes of quaternion algebras over F which split over E. For D ∈ X(E : F ), let JL = JL D denote the Jacquet-Langlands correspondence of representations from an inner form GL(n, D) to GL(2n) defined by Badulescu [2] and Badulescu-Renard [3], and LJ D denote its inverse. For a cuspidal representation π of GL(2n, A), π E denotes the base change of π to GL(2n, A E ), and X(E : F : π) denotes the set of D ∈ X(E : F ) for which π D = LJ D (π) exists as a (necessarily cuspidal) representation of GL(n, D)(A).…”

Periods of residual automorphic forms

“…(See [1,32,34].) For any irreducible unitary generic representation σ of a general linear group there are σ 1 , σ 2 , .…”

“…The mixed truncation enjoys many good properties. Its upshot is a rapidly decreasing function on H(F )\H(A) 1 On the other hand, the factorization of the global Rankin-Selberg integral and a local functional equation reflect a generic uniqueness principle, which states that the space of H v -invariant forms on π v I v (s) is at most one dimensional for generic values of s, where I v (s) is the induced representation corresponding to the Eisenstein series. Though this is a local statement, its proof is parallel to the unfolding of the global integral and will imply that if π v does not admit the relevant model, then there is no nonzero H v -invariant linear form on π v I v (s) for s in general position, so that P H (ϕ E(φ, s)) is necessarily identically zero and we will get…”

Induction automorphe pour GL(n,C)

“…Vogan a classifié [23] les représentations unitaires irréductibles de GL(n, F ). Nous utilisons ici une présentation du résultat parallèle à la classification de Tadić pour les corps locaux non archimédiens [20] ; grâce à la preuve de la conjecture de Kirillov par Baruch [4], on peut d'ailleurs maintenant donner une démonstration analogue dans les cas archimédiens et non archimédiens [21] voir aussi la présentation de [3].…”