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

We continue our study of the tempered spectrum of quasi-split classical groups. Here we examine the case of the special orthogonal groups of odd dimension. While this is the last of the classical groups to be examined, it is the first for which our results address the tempered spectrum whose supercuspidal support is an arbitrary maximal parabolic subgroup, as we describe below. We continue to see the connection between poles of local Langlands L–functions, reducibility of parabolically induced from supercuspidal representations, and the theory of twisted endoscopy. The recent progress in automorphic transfer and the local Langlands conjecture allows us to get more precise results than in previous cases. In particular, we can show that poles of the local Rankin-product L–functions are determined by local components of automorphic transfer, and for the most interesting case of GL2n×SO2n+1, the pole should be given precisely by this data. That we can also resolve reducibility for GLk × SO2n+1, for all n and k, stands in contrast to earlier cases, where we needed some restrictions. We let M ≃ GLn × SO2m+1 be an arbitrary maximal Levi subgroup of G = SO2r+1, with m + n = r. The main object of study for us is the standard intertwining operators and their poles. If τ ′ ⊗ τ is an irreducible supercuspidal representation of M , then (when τ is generic) the poles of the intertwining operators are those of the product of two local L–functions,