  We strengthen the compatibility between local and global Langlands
correspondences for GL_{n} when n is even and l=p. Let L be a CM field and \Pi\
a cuspidal automorphic representation of GL_{n}(\mathbb{A}_{L}) which is
conjugate self-dual and regular algebraic. In this case, there is an l-adic
Galois representation associated to \Pi, which is known to be compatible with
local Langlands in almost all cases when l=p by recent work of Barnet-Lamb,
Gee, Geraghty and Taylor. The compatibility was proved only up to
semisimplification unless \Pi\ has Shin-regular weight. We extend the
compatibility to Frobenius semisimplification in all cases by identifying the
monodromy operator on the global side. To achieve this, we derive a
generalization of Mokrane's weight spectral sequence for log crystalline
cohomology.
