The Schwartz space for the $(k,a)$-generalized Fourier transform and the minimal representation of the conformal group

This paper studies an analog of the classical Schwartz space $\mathcal{S}({\mathbb{R}}^N)$ in the framework of $(k,a)$-deformed harmonic analysis associated with the $(k,a)$-generalized Fourier transform ${\mathcal{F}}_{k,a}$. Motivated by the observation that $\mathcal{S}({\mathbb{R}}^N)$ coincides with the space of smooth vectors for the Segal–Shale–Weil representation, we define the $(k,a)$-generalized Schwartz space ${\mathcal{S}}_{k,a}({\mathbb{R}}^N)$ as the space of smooth vectors for the unitary representation associated with ${\mathcal{F}}_{k,a}$. Since this definition is intrinsic to the representation, it follows immediately that ${\mathcal{S}}_{k,a}({\mathbb{R}}^N)$ is preserved by ${\mathcal{F}}_{k,a}$. As main results, we explicitly determine ${\mathcal{S}}_{k,a}({\mathbb{R}}^N)$ 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 ${\widetilde{SO}}_0(N+1,2)$ studied by Kobayashi–Mano.