Contraction of the $ \mathfrak{sl}_2 $-triple associated to the $ (k, a) $-generalized Fourier transform
Tatsuro Hikawa
TL;DR
This work analyzes the contraction, as $a\to0$, of the $\mathfrak{sl}_2$-triple $\mathfrak{g}_{k,a}$ associated to the $(k,a)$-generalized Fourier framework, showing that $\mathfrak{g}_{k,0}$ becomes a three-dimensional commutative algebra $\mathfrak{g}_{k,0}\cong\mathbb{R}^3$. It establishes a joint spectral decomposition on $L^2(\mathbb{R}^N,w_{k,0})$ that lifts to a unitary representation of $\mathbb{R}^3$, yielding a continuous spectrum and enabling construction of operator semigroups with generators in $\mathfrak{g}_{k,0}$; the central analytic result gives the integral kernel for $\exp(z|x|^2\Delta_k)$ as a radial- angular series $K_k(x,x';z)$ with angular kernels $P_k^{(m)}$. In low dimensions, these kernels admit closed forms via the theta function, linking Dunkl analysis to explicit heat-kernel-type expressions. Overall, the paper bridges minimal representation theory, Dunkl analysis, and spectral theory through Lie-algebra contractions and semigroup kernels.
Abstract
Ben 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}} \{\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 \lvert x \rvert^2 Δ_k) $ as an infinite series, which, in some low-dimensional cases, can be expressed in a closed form using the theta function.