The tempered spectrum of quasi-split classical groups III: The odd orthogonal groups

Abstract

We continue our study of the poles of local Langlands L-functions through the theory of induced from supercuspidal representations of quasi-split groups. Here we study the odd special orthogonal groups, and hence determine poles of Rankin product L-functions. The pole of the intertwining operator is determined in terms of the theory of orbital integrals. This gives a description of the poles in terms of twisted endoscopy, as in previous cases. We use the language of functorial transfer to give precise descrption of the pole in terms of the local components of the global transfer, which has now been established.