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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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[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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