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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.11349 |
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| _version_ | 1866929388059099136 |
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| author | Arends, Christian Bang-Jensen, Frederik Frahm, Jan |
| author_facet | Arends, Christian Bang-Jensen, Frederik Frahm, Jan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_11349 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Explicit Hilbert spaces for the unitary dual of rank one orthogonal groups and applications Arends, Christian Bang-Jensen, Frederik Frahm, Jan Representation Theory 22E45 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. |
| title | Explicit Hilbert spaces for the unitary dual of rank one orthogonal groups and applications |
| topic | Representation Theory 22E45 |
| url | https://arxiv.org/abs/2406.11349 |