Title: Local-global compatibility and the action of monodromy on nearby cycles
Description: Abstract In this thesis, we study the compatibility between local and global Langlands correspondences for GLn. 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 GLn over the adeles of L, which is conjugate self-dual and regular algebraic. Under these assumptions, there is an l-adic Galois representation Rl<\sub>(&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 is not equal to 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.