Local-Global Compatibility and the Action of Monodromy on nearby Cycles

Abstract

In this thesis, we study the compatibility between local and global Langlands correspondences for (GL_n). This generalizes the compatibility between local and global class field theory and is related to deep conjectures in algebraic geometry and harmonic analysis, such as the Ramanujan-Petersson conjecture and the weight monodromy conjecture. Let L be a CM field. We consider the case when (\Pi) is a cuspidal automorphic representation of (GL_n(\mathbb{A}_L^\infty)), which is conjugate self-dual and regular algebraic. Under these assumptions, there is an l-adic Galois representation (R_l(\Pi)) associated to (\Pi), which is known to be compatible with the local Langlands correspondence in most cases (for example, when n is odd) and up to semisimplification in general. In this thesis, we complete the proof of the compatibility when (l \neq p) by identifying the monodromy operator N on both the local and the global sides. On the local side, the identification amounts to proving the Ramanujan-Petersson conjecture for (\Pi) as above. On the global side it amounts to proving the weight-monodromy conjecture for part of the cohomology of a certain Shimura variety.