The exterior square $L$-function on $\mathrm{GU}(2,2)$

TL;DR

This paper develops new integral representations for the exterior square L-function on the quasisplit unitary group GU(2,2) and related GSpin(6), advancing the understanding of automorphic L-functions through Rankin-Selberg integrals.

Contribution

It introduces novel Rankin-Selberg integrals for GU(2,2) and GSpin(6), providing explicit constructions for the exterior square L-function in these contexts.

Findings

Derived a two-variable integral for GU(2,2) involving cusp forms.

Reinterpreted GSpin(6) integrals originally studied by Gritsenko.

Analyzed the unfolding and non-uniqueness of the integral models.

Abstract

In this paper we give Rankin-Selberg integrals for the quasisplit unitary group on four variables, $\mathrm{GU}(2,2)$, and a closely-related quasisplit form of $\mathrm{GSpin}_6$. First, we give a two-variable Rankin-Selberg integral on $\mathrm{GU}(2,2)$. This integral applies to generic cusp forms, and represents the product of the exterior square (degree six) $L$-function and the standard (degree eight) $L$-function. Then we give a set of integral representations for just the degree six $L$-function on the quasisplit $\mathrm{GSpin}_6$. The $\mathrm{GSpin}_6$ integrals are reinterpretations of an integral originally considered by Gritsenko for Hermitian modular forms. We show that they unfold to a model that is not unique, and analyze the integrals via the technique of Piatetski-Shapiro and Rallis.