2018/09/24 by Shiv Prakash Patel, Patel, Shiv Prakash, Pooja Singla +1 · 1 citation
Mathematics · #15B33 #20C15 #20G05 #20G25 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1809.08743
openalex publication_date 2018/09/24 · openalex created_date 2019/02/21 · openalex updated_date 2026/07/28
Let \mathfrako 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 G be the General Linear or Special Linear group with entries from the finite quotients \mathfrako/\wp^ℓ of \mathfrako and U be the subgroup of G consisting of upper triangular unipotent matrices. We prove that the induced representation IndGU(θ) of G obtained from a \it 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 IndGU(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 over \mathfrakO/\wp^ℓ to the Special Linear groups is multiplicity free for all ℓ ≥ 2 and also obtain the corresponding branching rules in many cases.