LAUSR

We establish an explicit Plancherel decomposition for $${{\,\mathrm{\mathrm {GL}}\,}}_n(F){\backslash } {{\,\mathrm{\mathrm {GL}}\,}}_n(E)$$ where E/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 Zhang (J Am Math Soc 27:541–612, 2014) and secondly to the Hiraga–Ichino–Ikeda conjecture on formal degrees (J Am Math Soc 21(1):283–304, 2008) in the case of unitary groups.