Multiplicity one theorem over characteristic 2

TL;DR

This paper extends the multiplicity one theorem, originally proven for fields of characteristic not 2, to fields of characteristic 2 by adapting existing proof techniques.

Contribution

It provides the first proof of the multiplicity one theorem over characteristic 2 fields, filling a gap in the existing literature.

Findings

Distribution invariance under conjugation implies transposition invariance in characteristic 2

Adaptation of proof techniques to characteristic 2 fields

Extension of multiplicity one theorem to new field characteristic

Abstract

It is shown for all local fields $\mathbb{F}$ which are of characteristic different from $2$ that any distribution on $GL_{n+1}(\mathbb{F})$ which is invariant under conjugation by $GL_n(\mathbb{F})$ is also invariant under transposition. In this paper we give an adaptation of the proof of this theorem to fields of characteristic 2.