A two variable Rankin–Selberg integral for \({\textrm{GU}}(2,2)\) and the degree 5 L-function of \({\textrm{GSp}}_4\)

We give a two-variable Rankin–Selberg integral for generic cusp forms on \(\textrm{PGL}_4\) ${\text{PGL}}_4$ and \(\textrm{PGU}_{2,2}\) ${\text{PGU}}_{2,2}$ which represents a product of exterior square L-functions. As a residue of our integral, we obtain an integral representation on \(\textrm{PGU}_{2,2}\) ${\text{PGU}}_{2,2}$ of the degree 5 L-function of \({\textrm{GSp}}_4\) ${\text{GSp}}_4$ twisted by the quadratic character of E/F of cuspidal automorphic representations which contribute to the theta correspondence for the pair \((\textrm{P}{\textrm{GSp}}_4,\textrm{P}{\textrm{GU}}_{2,2})\) $(\text{P}{\text{GSp}}_4,\text{P}{\text{GU}}_{2,2})$ .