Multiplicity one theorems over positive characteristic

TL;DR

This paper proves multiplicity one theorems for classical groups over local fields of positive odd characteristic, establishing uniqueness of certain models and extending known results from characteristic zero.

Contribution

It extends multiplicity one theorems to orthogonal, unitary, and special orthogonal groups over positive characteristic fields, previously known only for general linear groups.

Findings

Proves multiplicity one for $O$, $U$, and $SO$ over positive odd characteristic fields.

Shows the uniqueness of Bessel, Rankin-Selberg, and Fourier-Jacobi models in this setting.

Utilizes the Gelfand-Kazhdan criterion to relate invariant distributions and anti-involutions.

Abstract

In [AGRS] a multiplicity one theorem is proven for general linear groups, orthogonal groups and unitary groups ($GL, O,$ and $U$) over $p$-adic local fields. That is to say that when we have a pair of such groups $G_n\subseteq G_{n+1}$, any restriction of an irreducible smooth representation of $G_{n+1}$ to $G_n$ is multiplicity free. This property is already known for $GL$ over a local field of positive characteristic, and in this paper we also give a proof for $O,U$, and $SO$ over local fields of positive odd characteristic. These theorems are shown in [GGP] to imply the uniqueness of Bessel models, and in [CS] to imply the uniqueness of Rankin-Selberg models. We also prove simultaniously the uniqeuness of Fourier-Jacobi models, following the outlines of the proof in [Sun]. By the Gelfand-Kazhdan criterion, the multiplicity one property for a pair $H\leq G$ follows from the statement that any distribution on $G$ invariant to conjugations by $H$ is also invariant to some anti-involution of $G$ preserving $H$.