A MULTIPLICITY ONE THEOREM FOR GROUPS OF TYPE A_n OVER DISCRETE VALUATION RINGS

Let 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 be the subgroup of G consisting of upper triangular unipotent matrices. We prove that the induced representation Ind^G_U(θ) of G obtained from a non-degenerate character θ of U is multiplicity free for all ℓ ≥ 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 are characterized by the property that these are the constituents of the induced representation Ind^G_U(θ) for some non-degenerate character θ of 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.