Residual spectrum of odd orthogonal groups

The purpose of this paper is fourfold. First, we show that the Langlands' functorial lift of a generic cuspidal representation of SO 2n + 1 to GL 2n is of the form $\textbf{Ind}{\boldsymbol\sigma}_\mathbf1\boldsymbol\otimes\boldsymbol\dots\boldsymbol\otimes{\boldsymbol\sigma}_\mathbf k$, where σ i 's are (unitary) cuspidal representations of ${\textbf{GL}}_{\mathbf2{\mathbf n}_\mathbf i}$ such that $\boldsymbol L{\boldsymbol(\mathbf s\boldsymbol,{\mathbf\sigma}_\mathbf i\boldsymbol,\mathbf\Lambda^\mathbf2\boldsymbol)}$, strengthening the recent result of Cogdell, Kim, Piatetski-Shapiro, and Shahidi. It uses the result of Barbasch and Moy on the spherical unitary dual,and the result of Ginzburg, Rallis, and Soudry that says that if $\boldsymbol L{\boldsymbol(\mathbf s\boldsymbol,\mathbf\sigma\boldsymbol\times\mathbf t\boldsymbol)}$, where $\boldsymbol\sigma$ is a cuspidal representation of ${\textbf{GL}}_\mathbf m$ and t is a generic cuspidal representation of ${\textbf{SO}}_{\mathbf2\mathbf n\boldsymbol+\mathbf1}$, has a pole at $\boldsymbol s\boldsymbol=\mathbf1$, then $\boldsymbol L{\boldsymbol(\mathbf s\boldsymbol,\mathbf\sigma\boldsymbol,\mathbf\Lambda^\mathbf2\boldsymbol)}$ has a pole at $\boldsymbol s\boldsymbol=\mathbf1$. Second, we give an application of the above result to the problem of determining the residual spectrum of ${\textbf{SO}}_{\mathbf2\mathbf n\boldsymbol+\mathbf1}$ coming from maximal parabolic subgroups,with the assumption that the completed symmetric square L-function of ${\textbf{GL}}_\mathbf n$ is holomorphic except possibly at $\boldsymbol s\boldsymbol=\mathbf0\boldsymbol,\mathbf1$. Third, we describe the residual spectrum of ${\textbf{SO}}_\mathbf5$ and ${\textbf{GSp}}_\mathbf4$, defined over a number field. Even though ${\textbf{Sp}}_\mathbf4$ and ${\textbf{SO}}_\mathbf5$ are closely related (isogenous), their residual spectrums are quite different. Fourth, we give a conjectural description of the residual spectrum of ${\textbf{SO}}_{\mathbf2\mathbf n\boldsymbol+\mathbf1}$ coming from arbitrary parabolic subgroups, assuming some standard global and local conjectures.