The local Langlands correspondence is a relationship between representations of the Galois group of a p-adic field F and the rerepresentations of GL_n(F).  Understanding the behavior of the local Langlands correspondence as one  varies Galois representations in families is an important ingredient in  Emerton's recent proof of many cases of the Fontaine-Mazur conjecture. I will explain this question, and its connection to questions involving the Bernstein center, an algebra that acts naturally on a category of representations of GL_n(F).