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

Abstract

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 $a\ne 0$. In the present paper, we study the behavior as the parameter $a$ approaches $0$. In this limit, the Lie algebra ${\mathfrak{g}}_{k,a}={\mathrm{span}}_{\mathbb{R}}\{{\mathbb{H}}_{k,a},{\mathbb{E}}_{k,a}^{+},{\mathbb{E}}_{k,a}^{-}\}\cong \mathfrak{sl}(2,\mathbb{R})$ 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 $\exp (z|x{|}^2{\Delta }_k)$ as an infinite series, which, in some low-dimensional cases, can be expressed in a closed form using the theta function.