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The sign of Galois representations attached to automorphic forms for unitary groups. (English) Zbl 1259.11058
Summary: 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 $$\pm 1$$, generalizing the fact that a self-dual absolutely irreducible representation is either symplectic or orthogonal. If $$\Pi$$ is a regular algebraic, polarized, cuspidal automorphic representation of $$\mathrm{GL}_{n}(\mathbb A_{K})$$, and if $$\rho$$ is a $$p$$-adic Galois representation attached to $$\Pi$$, 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 automorphic representations of $$\mathrm{GL}_{n}(\mathbb A_{F})$$ when $$F$$ is a totally real number field.

MSC:
 11F80 Galois representations 11F55 Other groups and their modular and automorphic forms (several variables)
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References:
 [2] doi:10.1515/crll.2004.031 · Zbl 1093.11036 [4] doi:10.4007/annals.2011.173.3.9 · Zbl 1269.11053 [9] doi:10.1007/s00222-005-0448-x · Zbl 1090.22008
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