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 by Harris-Taylor. More precisely, for the n-dimensional l-adic representation R_l(\Pi) of the Galois group of a CM-field L attached to a conjugate self-dual regular algebraic cuspidal automorphic representation \Pi, 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 prime v of L not dividing l corresponds to \Pi_v by the local Langlands correspondence. As a corollary, we show the irreducibility of R_l(\Pi) when \Pi_v is square integrable for some finite place v outside l. We also obtain conditional results in the case v|l.