# Multiplicity one for pairs of Prasad--Takloo-Bighash type

## TL;DR

This paper proves that for certain pairs of groups arising from central simple algebras over local fields, the multiplicity of distinguished representations is at most one, establishing a Gelfand pair property and self-duality of these representations.

## Contribution

It extends multiplicity one results to non-split cases over fields of positive characteristic and to archimedean fields, using Galois descent and global-to-local techniques.

## Key findings

- All double cosets are stable under inversion.
- Distinguished irreducible representations are self-dual.
- The pair (G,H) is a Gelfand pair with multiplicity at most one.

## Abstract

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