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

TL;DR

This paper extends integral representations of the exterior cube L-function from GL(6) to GU(6), analyzes its poles, introduces automorphic induction for GU(n), and proposes a criterion for pole existence based on Langlands functoriality.

Contribution

It develops a new integral representation for the exterior cube L-function on GU(6), introduces automorphic induction for GU(n), and links pole behavior to Langlands functoriality.

Findings

Extended integral representation to GU(6).

Established analytic properties and pole criteria for the L-function.

Proposed a conjectural criterion for poles based on automorphic induction.

Abstract

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.