The Gan-Gross-Prasad period of Klingen Eisenstein families over unitary groups

In this article, we compute the Gan-Gross-Prasad period integral of Klingen Eisenstein series over the unitary group $\mathrm{U}(m+1,n+1)$ with a cuspidal automorphic form over $\mathrm{U}(m+1,n)$, and show that it is related to certain special Rankin-Selberg $L$-values. We $p$-adically interpolate these Gan-Gross-Prasad period integrals as the Klingen Eisenstein series and the cuspidal automorphic form vary in Hida families. As a byproduct, we obtain a $p$-adic $L$-function of Rankin-Selberg type over $\mathrm{U}(m,n)\times \mathrm{U}(m+1,n)$. The ultimate motivation is to show the $p$-primitive property of Klingen Eisenstein series over unitary groups, by computing such Gan-Gross-Prasad period integrals, and this article is a starting point of this project. The $p$-primitivity of Eisenstein series is an essential property in the automorphic method in Iwasawa theory.