A Generalized Fourier Transform and a Smooth Analogue of Dunkl Operators
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915919978037248 |
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| author | Aoyama, Temma |
| author_facet | Aoyama, Temma |
| contents | We introduce a deformation of the Fourier transform on $\mathbb{R}^N$ arising from a representation-theoretic construction associated with $\widetilde{SL}(2,\mathbb{R}) \times O(N)$ that still admits an underlying degree-one operator structure. More precisely, we construct a generalized Fourier transform $\mathcal{F}_b$, a non-local deformation $H_b$ of the Laplacian $Δ$, and operators $D_{b,n}$ deforming the partial derivatives $\frac{\partial}{\partial x_n}$. We show that the operators $D_{b,n}$ and $x_n$ are compatible with the $\widetilde{SL}(2,\mathbb{R})$-representation in a way parallel to the classical case: for each $n$, the space spanned by $x_n$ and $D_{b,n}$ carries the standard representation of $\widetilde{SL}(2,\mathbb{R})$; in particular, the generalized Fourier transform $\mathcal{F}_b$ interchanges $D_{b,n}$ and $x_n$, and the $\mathfrak{sl}_2$-triple is recovered from quadratic expressions in these operators. We also establish the inversion formula for $\mathcal{F}_b$ and give explicit formulas for both $\mathcal{F}_b$ and $D_{b,n}$. In particular, $\mathcal{F}_b$ admits an explicit integral kernel representation, and $D_{b,n}$ is expressed as the sum of a differential term and a spherical integral term. Our construction might be viewed as a continuous analogue of Dunkl theory, with $O(N)$ playing the role of a reflection group. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_05421 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Generalized Fourier Transform and a Smooth Analogue of Dunkl Operators Aoyama, Temma Representation Theory Classical Analysis and ODEs Functional Analysis We introduce a deformation of the Fourier transform on $\mathbb{R}^N$ arising from a representation-theoretic construction associated with $\widetilde{SL}(2,\mathbb{R}) \times O(N)$ that still admits an underlying degree-one operator structure. More precisely, we construct a generalized Fourier transform $\mathcal{F}_b$, a non-local deformation $H_b$ of the Laplacian $Δ$, and operators $D_{b,n}$ deforming the partial derivatives $\frac{\partial}{\partial x_n}$. We show that the operators $D_{b,n}$ and $x_n$ are compatible with the $\widetilde{SL}(2,\mathbb{R})$-representation in a way parallel to the classical case: for each $n$, the space spanned by $x_n$ and $D_{b,n}$ carries the standard representation of $\widetilde{SL}(2,\mathbb{R})$; in particular, the generalized Fourier transform $\mathcal{F}_b$ interchanges $D_{b,n}$ and $x_n$, and the $\mathfrak{sl}_2$-triple is recovered from quadratic expressions in these operators. We also establish the inversion formula for $\mathcal{F}_b$ and give explicit formulas for both $\mathcal{F}_b$ and $D_{b,n}$. In particular, $\mathcal{F}_b$ admits an explicit integral kernel representation, and $D_{b,n}$ is expressed as the sum of a differential term and a spherical integral term. Our construction might be viewed as a continuous analogue of Dunkl theory, with $O(N)$ playing the role of a reflection group. |
| title | A Generalized Fourier Transform and a Smooth Analogue of Dunkl Operators |
| topic | Representation Theory Classical Analysis and ODEs Functional Analysis |
| url | https://arxiv.org/abs/2604.05421 |