Abstract.
We study the tempered spectrum of special orthogonal groups of odd degree over p-adic fields of characteristic zero. Residues of intertwining operators for representations parabolically induced from arbitrary maximal parabolic subgroups are determined in terms of non-vanishing of weighted sums of twisted orbital integrals. This is connected with the theory of twisted endoscopy, and poles of the Langlands L-functions, in particular, with the Rankin product L-function between irreducible (generic) supercuspidal representations of the classical group SO2m+1(F) and those of GLk(F). These poles are described for, any m and k, explicitly in terms of twisted endoscopy. Moreover, we show these poles describe precisely when a supercuspidal representation of GLk(F) is the local component of the transfer from a special odd orthogonal group.
Funding source: NSF
Award Identifier / Grant number: DMS9801340
Funding source: NSF
Award Identifier / Grant number: DMS0638799
Funding source: NSF
Award Identifier / Grant number: DMS0200325
Funding source: NSF
Award Identifier / Grant number: DMS0700280
Funding source: Guggenheim Fellowship
The authors would like to thank the Centre International de Recontres Mathématiques, in Luminy, France, and the Park City Mathematics Institute, where much of this manuscript was written. Both institutes provided pleasant environs in which to work and a high level of interesting and motivating mathematical activity.
© 2014 by Walter de Gruyter Berlin/Boston
