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

The $(k,a)$-generalized Fourier transform ${\mathcal{F}}_{k,a}$ 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 $a<0$.