Square root $p$-adic $L$-functions, I: Construction of a one-variable measure

The Ichino-Ikeda conjecture, and its generalization to unitary groups by N. Harris, has given explicit formulas for central critical values of a large class of Rankin-Selberg tensor products. Although the conjecture is not proved in full generality, there has been considerable progress, especially for $L$-values of the form $L(1\mathrm{/}2,BC(\pi )\times BC({\pi }^{\mathrm{\prime }}))$, where $\pi$ and ${\pi }^{\mathrm{\prime }}$ are cohomological automorphic representations of unitary groups $U(V)$ and $U(V^{\mathrm{\prime }})$, respectively. Here $V$ and $V^{\mathrm{\prime }}$ are hermitian spaces over a CM field, $V$ of dimension $n$, $V^{\mathrm{\prime }}$ of codimension $1$ in $V$, and $BC$ denotes the twisted base change to $GL(n)\times GL(n-1)$. This paper contains the first steps toward generalizing the construction of my paper with Tilouine on triple product $L$-functions to this situation. We assume $\pi$ is a holomorphic representation and ${\pi }^{\mathrm{\prime }}$ varies in an ordinary Hida family (of antiholomorphic forms). The construction of the measure attached to $\pi$ uses recent work of Eischen, Fintzen, Mantovan, and Varma.