Cherednik–Opdam transform or the hypergeometric transform. Its restriction to $W$-invariant functions is the transform that Heckman and Opdam studied, hence the name. A convenient analogy is to think of the Heckman–Opdam transform as a root space generalization of the spherical Fourier transform associated with a Riemannian symmetric space. The more general Cherednik–Opdam transform is not related to the Helgason–Fourier transform on a symmetric space, however.

The ${\mathbf{c}}$-function associated with $({\mathfrak{a}},\Sigma,m)$ is defined by $${\mathbf{c}}(\lambda)=c\prod_{\alpha\in\Sigma_0^+}{\mathbf{c}}_\alpha(\lambda),\quad \text{with}\quad {\mathbf{c}}_\alpha(\lambda)=\frac{2^{-\lambda_\alpha}}{\Gamma(\frac{\lambda_\alpha}{2}+\frac{m_\alpha}{4}+\frac{1}{2})\Gamma(\frac{\lambda_\alpha}{2}+\frac{m_\alpha}{4}+\frac{m_{2\alpha}}{2})},$$ where $c$ is a normalizing constant that is chosen so that ${\mathbf{c}}(\rho)=1$.

It is known that $|{\mathbf{c}}_\alpha(\lambda)|^{-2}\asymp|\!\left<\lambda,\alpha\right>\!|(1+\|\lambda\|)^{m_\alpha+m_{2\alpha}-2}$, so by the Cauchy–Schwarz inequality in ${\mathfrak{a}}$, $$\label{eqn.c-est1}
|{\mathbf{c}}_\alpha(\lambda)|^{-2}\asymp \begin{cases} \|\lambda\|^{2}(1+\|\lambda\|)^{m_\alpha+m_{2\alpha}-2}&\text{for }\lambda\in i{\mathfrak{a}}^*\text{ with } \|\lambda\| \text{ large},\\
\|\lambda\|^{2}&\text{for } \lambda\in i{\mathfrak{a}}^*\text{ with } \|\lambda\|\lesssim 1\end{cases}.$$ In particular, $$|{\mathbf{c}}(\lambda)|^{-2}\asymp \prod_{\alpha\in\Sigma_0^+}|\!\left<\lambda,\alpha\right>\!|^2(1+|\!\left<\lambda,\alpha\right>\!|)^{m_\alpha+m_{2\alpha}-2} \asymp \|\lambda\|^{2|\Sigma_0^+|}(1+\|\lambda\|)^{\beta-2|\Sigma_0^+|},$$ where $\beta=\sum_{\alpha\in\Sigma_0^+}(m_\alpha+m_{2\alpha})$ and where $|\Sigma_0^+|$ is the cardinality of $\Sigma_0^+$.

Let $dx$ denote a fixed normalization of the Haar measure on the abelian group ${\mathfrak{a}}$, and associate to $({\mathfrak{a}},\Sigma,m)$ the weighted measure $d\mu(x)=J(x)\,dx$ on ${\mathfrak{a}}$, where $$J(x)=\prod_{\alpha\in\Sigma^+}|e^{\alpha(x)}-e^{-\alpha(x)}|^{m_\alpha}.$$

It is known, cf. [@Narayanan-Pasquale-Pusti Theorem 1.13] and the references to the literature, that there exists a suitable normalization of the measure $d\lambda$ on $i{\mathfrak{a}}^*$ such that the transform $\mathcal{F}$ extends to an isometric isomorphism from $L^2({\mathfrak{a}},d\mu)^W$ onto 