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

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.

Final version. To appear in Canad. Math. Bull