The Schwartz space for the $ (k, a) $-generalized Fourier transform and the minimal representation of the conformal group
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912985149079552 |
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| author | Hikawa, Tatsuro |
| author_facet | Hikawa, Tatsuro |
| contents | This paper studies an analog of the classical Schwartz space $ \mathscr{S}(\mathbb{R}^N) $ in the framework of $ (k, a) $-deformed harmonic analysis associated with the $ (k, a) $-generalized Fourier transform $ \mathscr{F}_{k, a} $. Motivated by the observation that $ \mathscr{S}(\mathbb{R}^N) $ coincides with the space of smooth vectors for the Segal--Shale--Weil representation, we define the $ (k, a) $-generalized Schwartz space $ \mathscr{S}_{k, a}(\mathbb{R}^N) $ as the space of smooth vectors for the unitary representation associated with $ \mathscr{F}_{k, a} $. Since this definition is intrinsic to the representation, it follows immediately that $ \mathscr{S}_{k, a}(\mathbb{R}^N) $ is preserved by $ \mathscr{F}_{k, a} $. As main results, we explicitly determine $ \mathscr{S}_{k, a}(\mathbb{R}^N) $ for $ N = 1 $, as well as for general $ N $ when $ k = 0 $ and $ a $ is rational. We also explicitly determine the space of smooth vectors for the $ L^2 $-model of the minimal representation of the conformal group $ \widetilde{\mathit{SO}}_0(N + 1, 2) $ studied by Kobayashi--Mano. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_26453 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Schwartz space for the $ (k, a) $-generalized Fourier transform and the minimal representation of the conformal group Hikawa, Tatsuro Representation Theory Classical Analysis and ODEs Functional Analysis 42B10, 43A32, 22E45, 22E46, 46E10 This paper studies an analog of the classical Schwartz space $ \mathscr{S}(\mathbb{R}^N) $ in the framework of $ (k, a) $-deformed harmonic analysis associated with the $ (k, a) $-generalized Fourier transform $ \mathscr{F}_{k, a} $. Motivated by the observation that $ \mathscr{S}(\mathbb{R}^N) $ coincides with the space of smooth vectors for the Segal--Shale--Weil representation, we define the $ (k, a) $-generalized Schwartz space $ \mathscr{S}_{k, a}(\mathbb{R}^N) $ as the space of smooth vectors for the unitary representation associated with $ \mathscr{F}_{k, a} $. Since this definition is intrinsic to the representation, it follows immediately that $ \mathscr{S}_{k, a}(\mathbb{R}^N) $ is preserved by $ \mathscr{F}_{k, a} $. As main results, we explicitly determine $ \mathscr{S}_{k, a}(\mathbb{R}^N) $ for $ N = 1 $, as well as for general $ N $ when $ k = 0 $ and $ a $ is rational. We also explicitly determine the space of smooth vectors for the $ L^2 $-model of the minimal representation of the conformal group $ \widetilde{\mathit{SO}}_0(N + 1, 2) $ studied by Kobayashi--Mano. |
| title | The Schwartz space for the $ (k, a) $-generalized Fourier transform and the minimal representation of the conformal group |
| topic | Representation Theory Classical Analysis and ODEs Functional Analysis 42B10, 43A32, 22E45, 22E46, 46E10 |
| url | https://arxiv.org/abs/2603.26453 |