On periods and $L$-functions for $\mathrm{GU}(2,2) \times \mathrm{GL}(2)$

The study of periods of automorphic forms is a key theme in the Langlands program and has become an important tool to tackle various problems in Number Theory and Arithmetic Geometry.  For instance, Waldspurger formula and its generalisations have created a fertile ground for numerous arithmetic applications. In recent years, the conjectures of Sakellaridis and Venkatesh (and then Ben-Zvi, Sakellaridis, and Venkatesh) in the context of spherical varieties has led to a deeper understanding of automorphic periods and their relation to special values of $L$-functions. In this talk, I present work in progress aimed at looking at certain non-spherical cases. Precisely, I will describe a new integral representation of the degree 12 "exterior square x standard" $L$-function on generic cusp forms on $\mathrm{GU}(2,2) \times \mathrm{GL}(2)$ (or $\mathrm{GL}(4) \times \mathrm{GL}(2)$) and how it can be used to relate the non-vanishing of its central value to a certain cohomological period.  If time permits, I will describe how the same strategy applies to the case of $\mathrm{GSp}(6) \times \mathrm{GL}(2)$. This is joint work with Armando Gutierrez Terradillos.