# Multiplicity one theorem for $(\mathrm{GL}_{n+1},\mathrm{GL}_n)$ over a local field of positive characteristic

## TL;DR

This paper proves a multiplicity one theorem for representations of _{n+1} and _n over positive characteristic local fields, showing invariance under transposition and multiplicity-free restrictions.

## Contribution

It establishes a new multiplicity one result for _{n+1} and _n over positive characteristic fields, extending known results to this setting.

## Key findings

- Distributions invariant under _n action are transposition invariant.
- Restrictions of irreducible smooth representations are multiplicity free.
- The result applies to fields of positive characteristic different from 2.

## Abstract

Let $\mathbb{F}$ be a non-archimedean local field of positive characteristic different from 2. We consider distributions on $\mathrm{GL}(n+1,\mathbb{F})$ which are invariant under the adjoint action of $\mathrm{GL}(n,\mathbb{F})$. We prove that any such distribution is invariant with respect to transposition. This implies that the restriction to $\mathrm{GL}(n,\mathbb{F})$ of any irreducible smooth representation of $\mathrm{GL}(n+1,\mathbb{F})$ is multiplicity free.