On local-global compatibility for cuspidal regular algebraic automorphic representations of GLn

We prove the compatibility of local and global Langlands correspondences for GLn up to semisimplification for the Galois representations constructed by Harris-Lan-Taylor-Thorne and Scholze. More precisely, let rp(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 GLn( AF). We show that the restriction of rp(pi) to the decomposition group of a place v[special characters omitted]p of F corresponds up to semisimplification to rec(pi v), the image of piv under the local Langlands correspondence. Furthermore, we can show that the monodromy of the associated Weil-Deligne representation of rp(pi)∥ GalFv is 'more nilpotent' than the monodromy of rec(pi v).