Regular supercuspidal representations

Jiu-Kang Yu has given a general construction of supercuspidal representations of tamely ramified reductive p-adic groups. We will show that most of these representations can be parameterized by conjugacy classes of pairs consisting of an elliptic maximal torus and a character of it, subject to a simple and explicit root-theoretic condition. We will then draw a remarkable parallel between the characters of these representations and the characters of discrete series representations of real reductive groups. Guided by this parallel we shall give an explicit construction of the local Langlands correspondence for these representations. Furthermore, we shall recognize all but one of the pieces of the Langlands-Shelstad endoscopic transfer factor as coming from the character formula of discrete series/supercuspidal representations. We are thus lead to expect that similar terms will be relevant in character identities occurring beyond the endoscopic case.