A simple proof of the Langlands conjectures for GL(n) over a p-adic field

Let F be a finite extension of Q(p). For each integer n greater than or equal to 1, WP construct a bijection from the set G(F)(0)(n) of isomorphism classes of irreducible degree n representations of the (absolute) Well group of F, onto the set A(F)(0)(n) of isomorphism classes of smooth irreducible supercuspidal representations of GL(n)(F). Those bijections preserve epsilon factors for pairs and hence we obtain a proof of the Langlands conjectures for GL(n), over F, which is more direct than Harris and Taylor's. Our approach is global, and analogous to the derivation of local class held theory from global class field theory. We start with a result of Kottwitz and Clozel on the good reduction of some Shimura varieties and we use a trick of Harris, who constructs non-Galois automorphic induction in certain cases.