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

@misc{ABF24,
  title={Explicit Hilbert spaces for the unitary dual of rank one orthogonal groups and applications},
  author={Christian Arends and Frederik Bang-Jensen and Jan Frahm},
  year={2024},
  eprint={2406.11349},
  archivePrefix={arXiv},
  primaryClass={math.RT},
  url={https://arxiv.org/abs/2406.11349},
}