We strengthen the local-global compatibility of Langlands correspondences for GL_n in the case when n is even and l ≠ p. Let L be a CM field, and let ∏ be a cuspidal automorphic representation of GL_n(A_L) which is conjugate self-dual. Assume that ∏_∞ is cohomological and not "slightly regular," as defined by Shin. In this case, Chenevier and Harris constructed an l-adic Galois representation R_l(∏) and proved the local-global compatibility up to semisimplification at primes v not dividing l. We extend this compatibility by showing that the Frobenius semisimplification of the restriction of Rl(∏) to the decomposition group at v corresponds to the image of ∏_v via the local Langlands correspondence. We follow the strategy of Taylor and Yoshida, where it was assumed that ∏ is square-integrable at a finite place. To make the argument work, we study the action of the monodromy operator N on the complex of nearby cycles on a scheme which is locally étale over a product of strictly semistable schemes and we derive a generalization of the weight spectral sequence in this case. We also prove the Ramanujan-Petersson conjecture for ∏ as above.
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