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

Saïd-Kobayashi-Ørsted 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 \Delta_k\bigr) $ as an infinite series, which, in some low-dimensional cases, can be expressed in a closed form using the theta function.