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We derive two-sided bounds for the Newton and Poisson kernels of the W-invariant Dunkl Laplacian in the geometric complex case when the multiplicity 
$k(\alpha )=1$
i.e., for flat complex symmetric spaces. For the invariant Dunkl–Poisson kernel 
$P^{W}(x,y)$
, the estimates are where the 
$$ \begin{align*} P^{W}(x,y)\asymp \frac{P^{\mathbf{R}^{d}}(x,y)}{\prod_{\alpha> 0 \ }|x-\sigma_{\alpha} y|^{2k(\alpha)}}, \end{align*} $$
$\alpha $
’s are the positive roots of a root system acting in 
$\mathbf {R}^{d}$
, the 
$\sigma _{\alpha }$
’s are the corresponding symmetries and 
$P^{\mathbf {R}^{d}}$
is the classical Poisson kernel in 
${\mathbf {R}^{d}}$
. Analogous bounds are proven for the Newton kernel when 
$d\ge 3$
.
The same estimates are derived in the rank one direct product case 
$\mathbb {Z}_{2}^{N}$
and conjectured for general W-invariant Dunkl processes.
As an application, we get a two-sided bound for the Poisson and Newton kernels of the classical Dyson Brownian motion and of the Brownian motions in any Weyl chamber.
