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Inspired by a construction of Bump, Friedberg, and Ginzburg of a two-variableintegral representation on $\mathrm{GSp}_4$ for the product of the standard andspin $L$-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 thethree fundamental representations of the Langlands $L$-group$\mathrm{SL}_4(\mathbf{C})$. The second integral, which is closely related, isa 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 6exterior square $L$-function.

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