  On $\mathbb R^N$ equipped with a normalized root system $R$, a multiplicity
function $k(\alpha) > 0$, and the associated measure
  $$ dw(\mathbf x)=\prod_{\alpha\in R}|\langle \mathbf
x,\alpha\rangle|^{k(\alpha)}\, d\mathbf x, $$ let $h_t(\mathbf x,\mathbf y)$
denote the heat kernel of the semigroup generated by the Dunkl Laplace operator
$\Delta_k$. Let $d(\mathbf x,\mathbf y)=\min_{\sigma\in G} \| \mathbf
x-\sigma(\mathbf y)\|$, where $G$ is the reflection group associated with $R$.
We derive the following upper and lower bounds for $h_t(\mathbf x,\mathbf y)$:
for all $c_l>1/4$ and $0<c_u<1/4$ there are constants $C_l,C_u>0$ such that $$
  C_{l}w(B(\mathbf{x},\sqrt{t}))^{-1}e^{-c_{l}\frac{d(\mathbf{x},\mathbf{y})^2}{t}}
\Lambda(\mathbf x,\mathbf y,t) \leq h_t(\mathbf{x},\mathbf{y}) \leq
C_{u}w(B(\mathbf{x},\sqrt{t}))^{-1}e^{-c_{u}\frac{d(\mathbf{x},\mathbf{y})^2}{t}}
\Lambda(\mathbf x,\mathbf y,t), $$ where $\Lambda(\mathbf x,\mathbf y,t)$ can
be expressed by means of some rational functions of $\| \mathbf
x-\sigma(\mathbf y)\|/\sqrt{t}$. An exact formula for $\Lambda(\mathbf
x,\mathbf y,t)$ is provided.
