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

Let \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 \mathbf {U} be the subgroup of \mathbf {G} consisting of upper triangular unipotent matrices. We prove that the induced representation \operatorname {Ind}^{\mathbf {G}}_{\mathbf {U}}(\theta) of \mathbf {G} obtained from a 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 \operatorname {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 to the Special Linear groups is multiplicity free and also obtain the corresponding branching rules in many cases. [ABSTRACT FROM AUTHOR]