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

Abstract

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 $E/F$ is a quadratic extension of $p$-adic fields, the main result is a scalar Paley-Wiener theorem for relative Bessel distributions on $\mathrm{GL}_n(F)\backslash \mathrm{GL}_n(E)$. These distributions are relative character of the form $$π\mapsto I_π(f), \ f\in C_c^\infty(\mathrm{GL}_n(E)),$$ as $π$ ranges over $\mathrm{GL}_n(F)$-distinguished irreducible tempered representations, and are constructed from a $\mathrm{GL}_n(F)$-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 $U(n)$.