Explicit Hilbert spaces for the unitary dual of rank one orthogonal groups and applications

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Abstract

We realize all irreducible unitary representations of the group \(\mathrm{SO}_0(n+1,1)\) on explicit Hilbert spaces of vector-valued \(L^2\)-functions on \(\mathbb{R}^n\setminus\{0\}\). The key ingredient in our construction is an explicit expression for the standard Knapp–Stein intertwining operators between arbitrary principal series representations in terms of the Euclidean Fourier transform on a maximal unipotent subgroup isomorphic to \(\mathbb{R}^n\).

As an application, we describe the space of Whittaker vectors on all irreducible Casselman–Wallach representations. Moreover, the new realizations of the irreducible unitary representations immediately reveal their decomposition into irreducible representations of a parabolic subgroup, thus providing a simple proof of a recent result of Liu–Oshima–Yu.

Citation

@article {ABF26,
    AUTHOR = {Arends, Christian and Bang-Jensen, Frederik and Frahm, Jan},
     TITLE = {Explicit {H}ilbert spaces for the unitary dual of rank one
              orthogonal groups and applications},
   JOURNAL = {Adv. Math.},
  FJOURNAL = {Advances in Mathematics},
    VOLUME = {491},
      YEAR = {2026},
     PAGES = {Paper No. 110868},
       URL = {https://doi.org/10.1016/j.aim.2026.110868},
}