The spherical-function growth conjecture for Weyl-invariant Dunkl analysis

Let $\mathfrak{a}=\mathbf{R}^d$ be the Cartan algebra, let $\mathfrak{a}^+$ be the open positive Weyl chamber for a positive-root system $\Sigma^+$, and let $\overline{\mathfrak{a}^+}$ be its closure. For a multiplicity function $k$ and the associated spherical function $\psi_\lambda$, write $\alpha(X)$ for the root pairing and let $e^X$ denote the corresponding group exponential.

Spherical-function growth conjecture. If $\lambda,X\in\overline{\mathfrak{a}^+}$, then

$$\psi_\lambda(e^X)\asymp\frac{e^{\lambda(X)}}{\prod_{\alpha>0}(1+\alpha(X)\\,\alpha(\lambda))^{k(\alpha)}}.$$

For the root system $A_n$ on $\mathbf{R}^d$, with $d\geq n$ and constant multiplicity $k(\alpha)=k>0$, this is

$$\psi_\lambda(e^X)\asymp\frac{e^{\lambda(X)}}{\prod_{i<j\leq n+1}(1+(x_i-x_j)(\lambda_i-\lambda_j))^k},\qquad \lambda,X\in\overline{\mathfrak{a}^+},$$

where the implicit constants depend only on $k$. This estimate describes the boundary and interior growth of spherical functions and is the main estimate used to derive the corresponding Dunkl heat-kernel bounds. The statement is resolved in the $A_n$ case by the paper's proof of the conjecture.