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

TL;DR

This paper proves a multiplicity one theorem for induced representations of groups of type A_n over discrete valuation rings, extending classical results and characterizing regular representations through induced characters.

Contribution

It establishes a multiplicity one property for induced representations from non-degenerate characters in groups over discrete valuation rings, and characterizes regular representations via these induced representations.

Findings

Induced representations are multiplicity free for all  

Regular representations are characterized by constituents of induced representations

Restriction of regular representations to special linear groups is multiplicity free

Abstract

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.