Multivariate Rankin-Selberg integrals on $\mathrm{GL}_4$ and $\mathrm{GU}(2,2)$

TL;DR

This paper develops new multivariate Rankin-Selberg integral representations for automorphic L-functions on $ ext{GL}_4$ and $ ext{GU}(2,2)$, extending previous constructions to higher rank groups and multiple L-functions.

Contribution

It introduces two novel multivariate integral formulas for automorphic L-functions on $ ext{GL}_4$ and $ ext{GU}(2,2)$, representing products of fundamental and exterior square L-functions.

Findings

Three-variable integral for $ ext{PGL}_4$ L-functions

Two-variable integral for $ ext{PGU}(2,2)$ L-functions

Extension of Bump-Friedberg-Ginzburg construction to higher groups

Abstract

Inspired by a construction of Bump, Friedberg, and Ginzburg of a two-variable integral representation on $\mathrm{GSp}_4$ for the product of the standard and spin $L$-functions, we give two similar multivariate integral representations. The first is a three-variable Rankin-Selberg integral for cusp forms on $\mathrm{PGL}_4$ representing the product of the $L$-functions attached to the three fundamental representations of the Langlands $L$-group $\mathrm{SL}_4(\mathbf{C})$. The second integral, which is closely related, is a two-variable Rankin-Selberg integral for cusp forms on $\mathrm{PGU}(2,2)$ representing the product of the degree 8 standard $L$-function and the degree 6 exterior square $L$-function.