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Local-global compatibility and the action of monodromy on nearby cycles. (English) Zbl 1405.22028
Summary: We strengthen the local-global compatibility of Langlands correspondences for \(\mathrm{GL}_{n}\) in the case when \(n\) is even and \(l\neq p\). Let \(L\) be a CM field, and let \(\Pi\) be a cuspidal automorphic representation of \(\mathrm{GL}_{n}(\mathbb{A}_L)\) which is conjugate self-dual. Assume that \(\Pi_{\infty}\) is cohomological and not “slightly regular,” as defined by Shin. In this case, G. Chenevier and M. Harris [Camb. J. Math. 1, No. 1, 53–73 (2013; Zbl 1310.11062)] 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 R. Taylor and T. Yoshida [J. Am. Math. Soc. 20, No. 2, 467–493 (2007; Zbl 1210.11118)], 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.

MSC:
22E57 Geometric Langlands program: representation-theoretic aspects
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F80 Galois representations
11R39 Langlands-Weil conjectures, nonabelian class field theory
14G35 Modular and Shimura varieties
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