Compositio Mathematica

Research Article

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

Joël Bellaïchea1 and Gaëtan Cheneviera2

a1 Brandeis University, 415 South Street, Waltham, MA 02454-9110, USA (email:

a2 C.N.R.S., Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau Cedex, France (email:


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.

(Received December 22 2009)

(Accepted October 28 2010)

(Online publication July 27 2011)

2010 Mathematics Subject Classification

  • 11F80;
  • 11F55;
  • 11F85


  • Galois representation;
  • automorphic form;
  • unitary group;
  • sign;
  • symplectic;
  • orthogonal;
  • eigenvariety;
  • endoscopy;
  • p-adic


We thank Laurent Clozel, Michael Harris, and Jean-Pierre Labesse for many useful conversations and for their constant support. This paper relies on the book project [GRFAbook], and we thank all its authors for having made it possible. During the elaboration and writing of this paper, Joël Bellaïche was supported by the NSF grant DMS 05-01023, and Gaëtan Chenevier was supported by the C.N.R.S.