Compatibility of Local and Global Langlands Correspondences

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).