## 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 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.

Original language | English |
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Pages (from-to) | 2309-2322 |

Number of pages | 14 |

Journal | Proceedings of the American Mathematical Society |

Volume | 150 |

Issue number | 6 |

DOIs | |

State | Published - 1 Jan 2022 |

Externally published | Yes |

## Keywords

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

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics

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