LOCAL - GLOBAL COMPATIBILITY FOR l

We prove the compatibility of the local and global Langlands correspondences at places dividing l for the l-adic Galois representations associated to regular algebraic conjugate self-dual cuspidal automorphic representations of GLn over an imaginary CM field, under the assumption that the automorphic representations have Iwahori-fixed vectors at places dividing l and have Shin-regular weight. 2000 Mathematics Subject Classification. 11F33. The second author was partially supported by NSF grant DMS-0841491, the third author was partially supported by NSF grant DMS-0635607 and the fourth author was partially supported by NSF grant DMS-0600716 and by the Oswald Veblen and Simonyi Funds at the IAS.