Uniform Integrality of critical values of the Rankin-Selberg $L$-function for ${\mathrm{G}\mathrm{L}}_n\times {\mathrm{G}\mathrm{L}}_{n-1}$

After introducing the notion of uniform integrality of critical values of the Rankin-Selberg $L$-functions for ${\mathrm{G}\mathrm{L}}_n\times {\mathrm{G}\mathrm{L}}_{n-1}$, we study it when the base field is totally imaginary. For this purpose, we adopt specific models of highest weight representations of the general linear groups, construct the Eichler-Shimura classes for ${\mathrm{G}\mathrm{L}}_n$ and ${\mathrm{G}\mathrm{L}}_{n-1}$ in an explicit manner, and then evaluate the cohomological cup product of them, by making the best use of Gel’fand-Tsetlin basis.