A New Construction of the Tame Local Langlands Correspondence for GL(n,F), n a prime

Abstract

In this thesis, we give a new construction of the tame local Langlands correspondence for GL(n,F), n a prime, where F is a non-archimedean local field of characteristic zero. The Local Langlands Correspondence for GL(n,F) has been proven recently by Henniart, Harris/Taylor. In the tame case, supercuspidal representations correspond to characters of elliptic tori, but the local Langlands correspondence is unnatural because it involves a twist by some character of the torus. Taking the cue from the theory of real groups, supercuspidal representations should instead be parameterized by characters of covers of tori. Stephen DeBacker has calculated the distribution characters of supercuspidal representations for GL(n,F), n prime, and they are written in terms of functions on elliptic tori. Over the reals, Harish-Chandra parameterized discrete series representations of real groups by describing their distribution characters restricted to compact tori. Those distribution characters are written down in terms of functions on a canonical double cover of real tori. We show that if one writes down a natural analogue of Harish-Chandra's distribution character for p-adic groups, then it is the distribution character of a unique supercuspidal representation of GL(n,F), where n is prime, away from the local character expansion. These results pave the way for a natural construction of the tame local Langlands correspondence for GL(n,F), n a prime. In particular, there is no need to introduce any character twists.