The global spherical-function estimate for complex root systems

Let $G$ be a complex semisimple Lie group with flat Riemannian symmetric space, Cartan subalgebra $oindent\mathfrak{a}$, positive roots $\alpha>0$, and positive Weyl chamber $\overline{\mathfrak{a}}^+$. For $\lambda,X\in\overline{\mathfrak{a}}^+$, let $\psi_\lambda(X)$ denote the spherical function and write $f\asymp g$ when the two quantities are bounded above and below by positive constants independent of the variables under consideration.

The global spherical-function estimate. On flat Riemannian symmetric spaces with complex group $G$, one has

$$\psi_\lambda(X)\asymp \frac{e^{\langle\lambda,X\rangle}}{\prod_{\alpha>0}(1+\alpha(\lambda)\alpha(X))},\qquad \lambda,X\in\overline{\mathfrak{a}}^+.$$

The estimate is intended to be universal simultaneously in $\lambda$ and $X$. It is known in the complex type $A_n$ setting treated in the paper, while the cited results and partial asymptotics suggest that it should hold for arbitrary complex root systems.