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Compatibility of local and global Langlands correspondences. (English) Zbl 1210.11118

Summary: We prove the compatibility of local and global Langlands correspondences for \(\text{GL}_n\), which was proved up to semisimplification in [M. Harris and R. Taylor, The geometry and cohomology of some simple Shimura varieties, Ann. Math. Stud. 151, Princeton: Princeton University Press (2001; Zbl 1036.11027)]. More precisely, for the \(n\)-dimensional \( l\)-adic representation \( R_l(\Pi)\) of the Galois group of an imaginary CM-field \( L\) attached to a conjugate self-dual regular algebraic cuspidal automorphic representation \( \Pi\) of \(\text{GL}_n(\mathbb{A}_L)\), which is square integrable at some finite place, we show that Frobenius semisimplification of the restriction of \( R_l(\Pi)\) to the decomposition group of a place \( v\) of \( L\) not dividing \( l\) corresponds to \( \Pi_v\) by the local Langlands correspondence. If \( \Pi_v\) is square integrable for some finite place \( v \mid l\) we deduce that \( R_l(\Pi)\) is irreducible. We also obtain conditional results in the case \( v\mid l\).

MSC:

11R39 Langlands-Weil conjectures, nonabelian class field theory
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F80 Galois representations
14G35 Modular and Shimura varieties

Citations:

Zbl 1036.11027

References:

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