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

TL;DR

This paper investigates the poles of local Langlands L-functions for odd orthogonal groups, using orbital integrals and functorial transfer to describe these poles precisely in the context of twisted endoscopy.

Contribution

It extends the analysis of poles of local L-functions to odd orthogonal groups, providing a detailed description via orbital integrals and functorial transfer methods.

Findings

Determined poles of Rankin product L-functions for odd orthogonal groups.

Connected poles to orbital integrals and twisted endoscopy.

Provided explicit descriptions using functorial transfer.

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.