Cohomology classes, periods, and special values of Rankin-Selberg $L$-functions

In this article, we give a cohomological interpretation of (a special case of) the integrals constructed by the second named author and Q. Zhang \cite{YanZhang2023} which represent the product of Rankin-Selberg $L$-functions of $\mathrm{GL}_n\times\mathrm{GL}_m$ and $\mathrm{GL}_n\times\mathrm{GL}_{n-m-1}$ for $m<n$. As an application, we prove an algebraicity result for the special values of certain $L$-functions. This work is a generalization of the algebraicity result of Raghuram for $\mathrm{GL}_n\times\mathrm{GL}_{n-1}$ \cite{Raghuram2010} in the special case $m=n-1$, and the results of Mahnkopf \cite{Mahnkopf1998, Mahnkopf2005} in the special case $m=n-2$.