A Generalized Fourier Transform and a Smooth Analogue of Dunkl Operators

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 $\Delta$, 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.