Compatibility of local and global Langlands correspondences

Abstract

We prove the compatibility of local and global Langlands correspondences for G L n 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 n -dimensional l l -adic representation R l ( Π ) R_l(\Pi ) of the Galois group of an imaginary CM-field L L attached to a conjugate self-dual regular algebraic cuspidal automorphic representation Π \Pi of G L n ( A L ) GL_n(\mathbb A_L) , which is square integrable at some finite place, we show that Frobenius semisimplification of the restriction of R l ( Π ) R_l(\Pi ) to the decomposition group of a place v v of L L not dividing l l corresponds to Π v \Pi _v by the local Langlands correspondence. If Π v \Pi _v is square integrable for some finite place v ⧸ | l v \not | l we deduce that R l ( Π ) R_l(\Pi ) is irreducible. We also obtain conditional results in the case v | l v|l .