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

Let K be a CM number field and G(K) its absolute Galois group. A representation of G(K) 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 G(K) have a sign +/- 1, generalizing the fact that a self-dual absolutely irreducible representation is either symplectic or orthogonal. If II is a regular algebraic, polarized, cuspidal automorphic representation of GL(n) (A(K)), and if rho is a p-adic Galois representation attached to II, then rho 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 utomorphic representations of GL(n) (A(F)) when F is a totally real number field.