The sharp estimate for the Weyl-invariant heat kernel in the complex case

Let $p_t^W(X,Y)$ be the $W$-invariant heat kernel on the flat symmetric space, where $d$ is the dimension, $t>0$, $X,Y$ lie in the closed positive Weyl chamber, and $\alpha$ ranges over the positive roots. Write $f\asymp g$ for two-sided bounds by constants independent of the variables.

The Weyl-invariant heat-kernel estimate. One has

$$p_t^W(X,Y)\asymp t^{-\frac d2}\frac{e^{-\frac{|X-Y|^2}{4t}}}{\prod_{\alpha>0}(t+\alpha(X)\alpha(Y))}.$$

This is presented as the heat-kernel formulation equivalent to the preceding spherical-function conjecture. The estimate is proved in the paper for the $A_n$ case, but its extension beyond that setting remains conjectural.