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

Shiv Prakash Patel, Pooja Singla

Research output: Contribution to journalArticlepeer-review

Abstract

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 IndGU(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 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 to the Special Linear groups is multiplicity free and also obtain the corresponding branching rules in many cases.

Original languageEnglish
Pages (from-to)2309-2322
Number of pages14
JournalProceedings of the American Mathematical Society
Volume150
Issue number6
DOIs
StatePublished - 1 Jan 2022
Externally publishedYes

Keywords

  • Gelfand-Graev model
  • Whittaker model
  • multiplicity one
  • regular representations

ASJC Scopus subject areas

  • Mathematics (all)
  • Applied Mathematics

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