Multiplicity One for Pairs of Prasad–Takloo-Bighash Type

Abstract

Abstract Let ${\textrm{E}}/{\textrm{F}}$ be a quadratic extension of non-archimedean local fields of characteristic different from $2$. Let ${\textrm{A}}$ be an ${\textrm{F}}$-central simple algebra of even dimension so that it contains ${\textrm{E}}$ as a subfield, set ${\textrm{G}}={\textrm{A}}^\times $ and ${\textrm{H}}$ for the centralizer of ${\textrm{E}}^\times $ in ${\textrm{G}}$. Using a Galois descent argument, we prove that all double cosets ${\textrm{H}} g {\textrm{H}}\subset{\textrm{G}}$ are stable under the anti-involution $g\mapsto g^{-1}$, reducing to Guo’s result for ${\textrm{F}}$-split ${\textrm{G}}$ [ 14], which we extend to fields of positive characteristic different from $2$. We then show, combining global and local results, that ${\textrm{H}}$-distinguished irreducible representations of ${\textrm{G}}$ are self-dual and this implies that $({\textrm{G}},{\textrm{H}})$ is a Gelfand pair $$\begin{equation*}\operatorname{dim}_{\mathbb{C}}(\operatorname{Hom}_{{\textrm{H}}}(\pi,\mathbb{C}))\leq 1\end{equation*}$$for all smooth irreducible representations $\pi $ of ${\textrm{G}}$. Finally we explain how to obtain the multiplicity one statement in the archimedean case using the criteria of Aizenbud and Gourevitch ([ 1]), and we then show self-duality of irreducible distinguished representations in the archimedean case too.