Local-global compatibility for regular algebraic cuspidal automorphic representations when l ≠ p

We prove the compatibility of local and global Langlands correspondences for GL(n) up to semisimplification for the Galois representations constructed by Harris-Lan-Taylor-Thorne [10] and Scholze [18]. More precisely, let r(p)(pi) denote an n-dimensional p-adic representation of the Galois group of a CM field F attached to a regular algebraic cuspidal automorphic representation pi of GL(n)(A(F)). We show that the restriction of r(p)(pi) to the decomposition group of a place v <does not divide> p of F corresponds up to semisimplification to rec(pi(v)), the image of pi(v) under the local Langlands correspondence. Furthermore, we can show that the monodromy of the associated Weil-Deligne representation of r(p)(pi)|(GalFv) is 'more nilpotent' than the monodromy of rec(pi(v)).