  Let $\Delta$ and $L=\Delta -\|\mathbf x\|^2$ be the Dunkl Laplacian and the
Dunkl harmonic oscillator respectively. We define the Hardy space $\mathcal
H^1$ associated with the Dunkl harmonic oscillator by means of the
nontangential maximal function with respect to the semigroup $e^{tL}$. We prove
that the space $\mathcal H^1$ admits characterizations by relevant Riesz
transforms and atomic decompositions. The atoms which occur in the atomic
decompositions are of local type.
