Wavelet transform and orthogonal decomposition of $L^{2}$ space on the Cartan domain $BDI(q=2)$

Let $G=\left ({\mathbb {R}}^{*}_{+}\times SO_{0}(1, n)\right ) \ltimes {\mathbb {R}}^{n+1}$ be the Weyl-Poincaré group and $KAN$ be the Iwasawa decomposition of $SO_{0}(1, n)$ with $K=SO(n)$. Then the ``affine Weyl-Poincaré group'' $G_{a}=\left ({\mathbb {R}}^{*}_{+}\times AN\right ) \ltimes {\mathbb {R}}^{n+1}$ can be realized as the complex tube domain $\Pi ={\mathbb {R}}^{n+1}+iC$ or the classical Cartan domain $BDI(q=2)$. The square-integrable representations of $G$ and $G_{a}$ give the admissible wavelets and wavelet transforms. An orthogonal basis $\{ \psi _{k}\}$ of the set of admissible wavelets associated to $G_{a}$ is constructed, and it gives an orthogonal decomposition of $L^{2}$ space on $\Pi$ (or the Cartan domain
$BDI(q=2)$) with every component $A_{k}$ being the range of wavelet transforms of functions in $H^{2}$ with $\psi _{k}$.