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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
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