Compatibility of local and global Langlands correspondences

Taylor, Richard; Yoshida, Teruyoshi

doi:10.1090/S0894-0347-06-00542-X

Compatibility of local and global Langlands correspondences
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by Richard Taylor and Teruyoshi Yoshida

J. Amer. Math. Soc. 20 (2007), 467-493

DOI: https://doi.org/10.1090/S0894-0347-06-00542-X

Published electronically: July 10, 2006

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Abstract:

We prove the compatibility of local and global Langlands correspondences for $GL_n$, which was proved up to semisimplification in M. Harris and R. Taylor, The Geometry and Cohomology of Some Simple Shimura Varieties, Ann. of Math. Studies 151, Princeton Univ. Press, Princeton-Oxford, 2001. 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 $GL_n(\mathbb A_L)$, 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 $v \not | l$ we deduce that $R_l(\Pi )$ is irreducible. We also obtain conditional results in the case $v|l$.

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