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

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