The Exterior Cubic L-function of GU(6) and Unitary Automorphic Induction

In this paper, we extend Ginzburg-Rallis' integral representation for the exterior cube automorphic $L$-function of ${\rm GL}_6\times {\rm GL}_1$ to that of the quasi-split unitary similitude group ${\rm GU}_6$ and establish its analytic properties to determine the poles of this $L$-function. Furthermore, we introduce the automorphic induction for automorphic representations of ${\rm GU}_n$ and then show that the weak Langlands functorial lift for the automorphic induction exists for generic cuspidal automorphic representations. By using this automorphic induction, we give a conjectural criterion on the existence of poles of $L(s,\pi,\wedge^{3}\otimes\chi)$ for discrete automorphic representations in the tempered spectrum.