in ${\mathfrak{a}}^*$ and let $W$ denote the associated Weyl group generated by the root reflections $r_\alpha:\lambda\mapsto\lambda -2\lambda_\alpha\alpha$ for $\alpha\in\Sigma$. Fix a compatible set $\Sigma^+$ of positive roots in $\Sigma$ and let $\Pi=
\{\alpha_1,\ldots,\alpha_n\}\subset\Sigma^+$ be the associated set of simple roots. Let $\Sigma_0$ denote the set of roots in $\Sigma$ that are indivisible, in the sense that if $\alpha$ belongs to $\Sigma_0$, then $\alpha/2$ is not a root. A (strictly) positive multiplicity function is a $W$-invariant function $m:\Sigma\to(0,\infty)$. We often write $m_\alpha=m(\alpha)$. By $W$-invariance it holds that $m_{w\alpha}=m_\alpha$ for all $\alpha\in\Sigma$ and $w\in W$. We adhere to the conventions in [@Narayanan-Pasquale-Pusti] rather than Heckman and Opdam: Their root system $\mathfrak{R}$ and multiplicity function $k$ are related to $\Sigma$ and $m$ above by the identities $\mathfrak{R}=\{2\alpha\,:\,\alpha\in\Sigma\}$, $k_{2\alpha}=m_\alpha/2$ for $\alpha\in\Sigma$. Set $\Sigma_0^+=\Sigma^+\cap\Sigma_0$. The complexification ${\mathfrak{a}}_{\mathbb{C}}$ of ${\mathfrak{a}}$ may be viewed as the Lie algebra of the complex torus $A_{\mathbb{C}}={\mathfrak{a}}_{\mathbb{C}}/\{2\pi ix_\alpha/\langle\alpha,\alpha\rangle\,:\,\alpha\in\Sigma\}{\mathbb{Z}}$. Let $\exp:{\mathfrak{a}}_{\mathbb{C}}\to A_{\mathbb{C}}$ be the exponential map. The real form $A=\exp{\mathfrak{a}}$ of $A_{\mathbb{C}}$ is an abelian subgroup of $A_{\mathbb{C}}$ with Lie algebra ${\mathfrak{a}}$ such that $\exp:{\mathfrak{a}}\to A$ is a diffeomorphism, by means of which we shall often identify ${\mathfrak{a}}$ with $A$. The $W$-action extends to ${\mathfrak{a}}$ by duality, and to ${\mathfrak{a}}_{\mathbb{C}}^*$ and ${\mathfrak{a}}_{\mathbb{C}}$ 