# Compatibility of Local and Global Langlands Correspondences

 Title: Compatibility of Local and Global Langlands Correspondences Author: Taylor, Richard L.; Yoshida, Teruyoshi Note: Order does not necessarily reflect citation order of authors. Citation: Taylor, Richard L., and Teruyoshi Yoshida. 2007. Compatibility of local and global langlands correspondences. Journal of the American Mathematical Society 20: 467-493. Full Text & Related Files: Taylor_CompatibilityLanglands.pdf (269.5Kb; PDF) 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 $$v \not\vert l$$ we deduce that $$R_l(\Pi)$$ is irreducible. We also obtain conditional results in the case of $$v\vert l$$. Published Version: doi:10.1090/S0894-0347-06-00542-X Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:3550505 Downloads of this work: