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The sign of Galois representations attached to automorphic forms for unitary groups

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  27 July 2011

Joël Bellaïche  and
Gaëtan Chenevier
Joël Bellaïche
Affiliation:
Brandeis University, 415 South Street, Waltham, MA 02454-9110, USA (email: jbellaic@brandeis.edu)
Gaëtan Chenevier
Affiliation:
C.N.R.S., Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau Cedex, France (email: chenevier@math.polytechnique.fr)
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Abstract

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Let K be a CM number field and GK its absolute Galois group. A representation of GK is said to be polarized if it is isomorphic to the contragredient of its outer complex conjugate, up to a twist by a power of the cyclotomic character. Absolutely irreducible polarized representations of GK have a sign ±1, generalizing the fact that a self-dual absolutely irreducible representation is either symplectic or orthogonal. If Π is a regular algebraic, polarized, cuspidal automorphic representation of GLn(𝔸K), and if ρ is a p-adic Galois representation attached to Π, then ρ is polarized and we show that all of its polarized irreducible constituents have sign +1 . In particular, we determine the orthogonal/symplectic alternative for the Galois representations associated to the regular algebraic, essentially self-dual, cuspidal automorphic representations of GLn (𝔸F) when F is a totally real number field.

Keywords

Galois representationautomorphic formunitary groupsignsymplecticorthogonaleigenvarietyendoscopyp-adic

MSC classification

Secondary: 11F80: Galois representations 11F55: Other groups and their modular and automorphic forms (several variables) 11F85: $p$-adic theory, local fields
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 5 , September 2011 , pp. 1337 - 1352
Copyright
Copyright © Foundation Compositio Mathematica 2011

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