Compatibility of local and global Langlands correspondences

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 $v\vert l$.