arXiv subjects

Tatsuro Hikawa

Publications and source records attributed to Tatsuro Hikawa.

3 recordsLinked to original sources

arXiv2026

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 $ \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.

math.RT↗

arXiv2026

Contraction of the $\mathfrak{sl}_2$-Triple Associated to the $(k,a)$-Generalized Fourier Transform

Ben Sa\"ıd-Kobayashi-Orsted introduced a family of $ \mathfrak{sl}_2 $-triples of differential-difference operators $ \mathbb{H}_{k,a} $, $ \mathbb{E}^+_{k,a} $ and $ \mathbb{E}^-_{k,a} $ on $ \mathbb{R}^N \setminus \{0\} $ indexed by a Dunkl parameter $ k $ and a deformation parameter $ a \neq 0 $. In the present paper, we study the behavior as the parameter $ a $ approaches $ 0 $. In this limit, the Lie algebra $ \mathfrak{g}_{k,a} = \operatorname{span}_\mathbb{R} \bigl\{\mathbb{H}_{k,a}, \mathbb{E}^+_{k,a}, \mathbb{E}^-_{k,a}\bigr\} \cong \mathfrak{sl}(2, \mathbb{R}) $ contracts to a three-dimensional commutative Lie algebra $ \mathfrak{g}_{k,0}$, and its spectral properties change. We describe the joint spectral decomposition for $ \mathfrak{g}_{k,0}$, and discuss formulas for operator semigroups with infinitesimal generators in $ \mathfrak{g}_{k,0}$. In particular, we describe the integral kernel of $ \exp\bigl(z |x|^2 Δ_k\bigr) $ as an infinite series, which, in some low-dimensional cases, can be expressed in a closed form using the theta function.

math.RT↗

arXiv2025

$(k, a)$-generalized Fourier transform with negative $a$

The $ (k, a) $-generalized Fourier transform $ \mathscr{F}_{k, a} $ introduced by Ben Saïd--Kobayashi--Ørsted is a deformation family of the classical Fourier transform with a Dunkl parameter $ k $ and a parameter $ a > 0 $ that interpolates minimal representations of two different simple Lie groups. In the present paper, we focus on the case $ a < 0 $. As a main result, we find a unitary transform that intertwines the known case $ a > 0 $ and the new case $ a < 0 $.

math.RT↗