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 \(v\vert l\).