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

Abstract

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.