A trace Paley-Wiener theorem for ${\mathrm{G}\mathrm{L}}_n(F)\mathrm{\backslash }{\mathrm{G}\mathrm{L}}_n(E)$, with $E\mathrm{/}F$ a quadratic extension of non-Archimedean local fields

We extend the classical trace Paley-Wiener theorem for $p$-adic reductive groups, given by Bernstein, Deligne and Kazhdan in 1986, to a relative case. That is when $E\mathrm{/}F$ is a quadratic extension of $p$-adic fields, the main result is a scalar Paley-Wiener theorem for relative Bessel distributions on ${\mathrm{G}\mathrm{L}}_n(F)\mathrm{\backslash }{\mathrm{G}\mathrm{L}}_n(E)$. These distributions are relative character of the form $$\pi \mapsto I_{\pi }(f),f\in C_c^{\mathrm{\infty }}({\mathrm{G}\mathrm{L}}_n(E)),$$ as $\pi$ ranges over ${\mathrm{G}\mathrm{L}}_n(F)$-distinguished irreducible tempered representations, and are constructed from a ${\mathrm{G}\mathrm{L}}_n(F)$-invariant functional and a Whittaker functional. Using the local Langlands correspondence, and the base-change from a unitary group, we show that the relative characters can be described as elements of the “generic” Bernstein center of the unitary group $U(n)$.