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Construction of automorphic Galois representations. II. (English) Zbl 1310.11062
Let \(F\) be a totally real field and \(K\) a totally imaginary quadratic extension of \(F\). This paper constructs Galois representations associated to certain cuspidal automorphic representations of \(\mathrm{GL}(n,K)\): Let \(\Pi\) be a cuspidal automorphic representation of \(\mathrm{GL}(n,K)\) that is cohomological and ‘conjugate self-dual’. Then there exists a semi-simple continuous Galois representation \(\rho_{\iota,\Pi} : G_K \rightarrow \mathrm{GL}(n,\overline{\mathbb Q}_p)\) associated to \(\Pi\) (where \(\iota\) is are choices of embeddings of \(\overline{\mathbb Q}\) into \(\overline{\mathbb Q}_p\) and \(\mathbb C\)) which satisfying some local compatibility conditions at places \(v\) not dividing \(p\) and as well as dividing \(p\). This main result is achieved by constructing missing Galois representations for even degree \(F\) by \(p\)-adic approximation and has applications to automorphy lifting theorems.
This is a revised version of the articles “Endoscopic transfer”, and “Construction of automorphic Galois representations. I” by L. Clozel, M. Harris and J.-P. Labesse [Stabilization of the trace formula, Shimura varieties, and arithmetic applications. Volume 1: On the stabilization of the trace formula. Somerville, MA: International Press, 475–496, 497–523 (2011; Zbl 1255.11027)] and of an earlier version of the present article [http://fa.institut.math.jussieu.fr/node/45].

MSC:
11F80 Galois representations
11F85 \(p\)-adic theory, local fields
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11S37 Langlands-Weil conjectures, nonabelian class field theory
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
22E50 Representations of Lie and linear algebraic groups over local fields
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