A multiplicity one theorem for groups of type 𝐴_{𝑛} over discrete valuation rings

Abstract

Let G \mathbf {G} be the General Linear or Special Linear group with entries from the finite quotients of the ring of integers of a non-archimedean local field and U \mathbf {U} be the subgroup of G \mathbf {G} consisting of upper triangular unipotent matrices. We prove that the induced representation Ind U G ⁡ ( θ ) \operatorname {Ind}^{\mathbf {G}}_{\mathbf {U}}(\theta ) of G \mathbf {G} obtained from a non-degenerate character θ \theta of U \mathbf {U} is multiplicity free for all ℓ ≥ 2. \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 G \mathbf {G} are characterized by the property that these are the constituents of the induced representation Ind U G ⁡ ( θ ) \operatorname {Ind}^{\mathbf {G}}_{\mathbf {U}}(\theta ) for some non-degenerate character θ \theta of U \mathbf {U} . We use this to prove that the restriction of a regular representation of General Linear groups to the Special Linear groups is multiplicity free and also obtain the corresponding branching rules in many cases.