A multiplicity one theorem for groups of type $A_n$ over discrete valuation rings

Let $\mathfrak{o}$ be the ring of integers of a non-archimedean local field with the maximal ideal $\wp$ and the finite residue field of characteristic $p.$ Let $\mathbf{G}$ be the General Linear or Special Linear group with entries from the finite quotients $\mathfrak{o}/\wp^\ell$ of $\mathfrak{o}$ and $\mathbf{U}$ be the subgroup of $\mathbf{G}$ consisting of upper triangular unipotent matrices. We prove that the induced representation $\mathrm{Ind}^{\mathbf{G}}_{\mathbf{U}}(\theta)$ of $\mathbf{G}$ obtained from a ${\it non-degenerate}$ character $\theta$ of $\mathbf{U}$ is multiplicity free for all $\ell \geq 2.$ This is analogous to the multiplicity one theorem regarding Gelfand-Graev representation for the finite Chevalley groups. We prove that for many cases the regular representations of $\mathbf{G}$ are characterized by the property that these are the constituents of the induced representation $\mathrm{Ind}^{\mathbf{G}}_{\mathbf{U}}(\theta)$ for some non-degenerate character $\theta$ of $\mathbf{U}$. We use this to prove that the restriction of a regular representation of General Linear groups over $\mathfrak{O}/\wp^\ell$ to the Special Linear groups is multiplicity free for all $\ell \geq 2$ and also obtain the corresponding branching rules in many cases.