Exterior-cubic pole criterion for tempered cuspidal representations of GU(6)

Let $F$ be the global field and let $E/F$ be the quadratic extension underlying the unitary similitude group. Let $\ba$ denote the ring of adeles of $F$, let $\pi$ be an irreducible cuspidal automorphic representation of $\operatorname{GU}_6(\ba)$ in the tempered discrete spectrum defined in the endoscopic classification conjecture, and let $\chi$ be the character occurring in the exterior-cubic $L$-function. Let $E^+/F$ be the quadratic extension associated with the non-trivial quadratic character $\omega_\pi\chi^2$, put $K=E\otimes_F E^+$, and let $\omega_\tau$ denote the central character of $\tau$. Exterior-cubic pole criterion. The partial $L$-function $L^S(s,\pi,\wedge^3\otimes\chi)$ has a pole at $s=1$ if and only if $\pi$ is an automorphic induction $i_{E^+/F,\xi}(\tau)$ from an irreducible cuspidal automorphic representation $\tau$ of $\operatorname{GU}^{\circ}_{K/E^+}(3,\ba_F)$ in the tempered discrete spectrum and $L(s,i_F(\omega_\tau)\otimes\chi)$ has a pole at $s=1$. This conjecture gives a criterion for poles of the exterior-cubic $L$-function in terms of automorphic induction; the paper verifies it for certain cuspidal representations arising from endoscopic lifting, while the general case is left for future work.