Local-global compatibility and the action of monodromy on nearby cycles

Abstract

We strengthen the local-global compatibility of Langlands correspondences for ${GL}_n$ in the case when $n$ is even and $l\ne p$. Let $L$ be a CM field, and let $\Pi$ be a cuspidal automorphic representation of ${GL}_n\left({\mathbb{A}}_L\right)$ which is conjugate self-dual. Assume that ${\Pi }_{\infty }$ is cohomological and not “slightly regular,” as defined by Shin. In this case, Chenevier and Harris constructed an $l$-adic Galois representation $R_l(\Pi )$ 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 $R_l(\Pi )$ to the decomposition group at $v$ corresponds to the image of ${\Pi }_v$ via the local Langlands correspondence. We follow the strategy of Taylor and Yoshida, where it was assumed that $\Pi$ 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 $\Pi$ as above.