Heisenberg parabolically induced representations of Hermitian Lie groups, Part I: Intertwining operators and Weyl transform

Jan Frahm*, Clemens Weiske, Genkai Zhang

*Corresponding author for this work

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1 Citation (Scopus)

Abstract

For a Hermitian Lie group G, we study the family of representations induced from a character of the maximal parabolic subgroup P=MAN whose unipotent radical N is a Heisenberg group. Realizing these representations in the non-compact picture on a space I(ν) of functions on the opposite unipotent radical N¯, we apply the Heisenberg group Fourier transform mapping functions on N¯ to operators on Fock spaces. The main result is an explicit expression for the Knapp–Stein intertwining operators I(ν)→I(−ν) on the Fourier transformed side. This gives a new construction of the complementary series and of certain unitarizable subrepresentations at points of reducibility. Further auxiliary results are a Bernstein–Sato identity for the Knapp–Stein kernel on N‾ and the decomposition of the metaplectic representation under the non-compact group M.

Original languageDanish
Article number109001
JournalAdvances in Mathematics
Volume422
Number of pages44
ISSN0001-8708
DOIs
Publication statusPublished - Apr 2023

Keywords

  • Complementary series
  • Heisenberg parabolic subgroups
  • Hermitian Lie groups
  • Induced representations
  • Unitarizable subrepresentations
  • 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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