(joint with Sug Woo Shin) We extend the methods of our proof of the LLC to general (possibly ramified) PEL data. In particular, we formulate a precise conjecture relating the cohomology of deformation spaces of p-divisible groups with PEL (or EL) structure to the Langlands correspondence, and prove this conjecture in all cases of EL type. Using the Langlands-Kottwitz method, one can deduce new results about the cohomology of compact unitary group Shimura varieties at ramified split places, and reprove results of Shin about the existence of Galois representations associated to RACSD cuspidal automorphic representations.