Plancherel formula for ${\mathrm{G}\mathrm{L}}_n(F)\mathrm{\backslash }{\mathrm{G}\mathrm{L}}_n(E)$ and applications to the Ichino-Ikeda and formal degree conjectures for unitary groups

We establish an explicit Plancherel decomposition for ${\mathrm{G}\mathrm{L}}_n(F)\mathrm{\backslash }{\mathrm{G}\mathrm{L}}_n(E)$ where $E\mathrm{/}F$ is a quadratic extension of local fields of characteristic zero by making use of a local functional equation for Asai $\gamma$-factors. We also give two applications of this Plancherel formula: first to the global Ichino-Ikeda conjecture for unitary groups by completing a comparison between local relative characters that was left open by W. Zhang and secondly to the Hiraga-Ichino-Ikeda conjecture on formal degrees in the case of unitary groups.