A multiplicity one theorem for groups of type 𝐴_{𝑛} over discrete valuation rings

Patel, Shiv; Singla, Pooja

doi:10.1090/proc/15816

A multiplicity one theorem for groups of type $A_n$ over discrete valuation rings
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by Shiv Prakash Patel and Pooja Singla

Proc. Amer. Math. Soc. 150 (2022), 2309-2322

DOI: https://doi.org/10.1090/proc/15816

Published electronically: March 16, 2022

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Abstract:

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.

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