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

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.