Symmetry breaking for PGL(2) over non-archimedean local fields

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Abstract

For a quadratic extension $\mathbb{E}/\mathbb{F}$ of non-archimedean local fields we construct explicit holomorphic families of intertwining operators between principal series representations of $\operatorname{PGL}(2,\mathbb{E})$ and $\operatorname{PGL}(2,\mathbb{F})$, also referred to as symmetry breaking operators. These families are given in terms of their distribution kernels which can be viewed as distributions on $\mathbb{E}$ depending holomorphically on the principal series parameters. For all such parameters we determine the support of these distributions, and we study their mapping properties. This leads to a classification of all intertwining operators between principal series representations, not necessarily irreducible. As an application, we show that every Steinberg representation of $\operatorname{PGL}(2,\mathbb{E})$ contains a Steinberg representation of $\operatorname{PGL}(2,\mathbb{F})$ as a direct summand of Hilbert spaces.
Original languageEnglish
Title of host publicationRepresentations and Characters: Revisiting the Works of Harish-Chandra and André Weil : IMS Lecture Note Series
Number of pages42
Publication statusAccepted/In press - 31 Jan 2024

Keywords

  • math.RT
  • Primary 22E35, Secondary 22E50

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  • Limits of p-adic geometries

    Ciobotaru, C.-G. (PI)

    01/06/202331/05/2028

    Project: Research

  • Symmetry Breaking in Mathematics

    Frahm, J. (PI), Weiske, C. (Participant), Ditlevsen, J. (Participant), Spilioti, P. (Participant), Bang-Jensen, F. J. (Participant) & Labriet, Q. (Participant)

    01/08/201931/07/2024

    Project: Research

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