Natural action on Hilbert Schemes

Question

I'm reading chapter 4 of Nakajima's Lectures on Hilbert Schemes of Points on Surfaces and I'm a little confused. He says the natural action of $\Gamma$ on $\mathbb{C}^2$ gives a natural action on the Hilbert Scheme $(\mathbb{C}^2)^{[N]}$ where $\Gamma$ is a finite subgroup of $SU(2)$ of order $N$. I am not sure what this action is. I see that there is a natural action on $S^N (\mathbb{C}^2)$, the Nth symmetric product, since that comprises sets of $N$ points. But I can't see the natural action on $(\mathbb{C}^2)^{[N]}$. Am I supposed to have in mind that $(\mathbb{C}^2)^{[N]}$ is a resolution of $S^N (\mathbb{C}^2)$? I imagine this is a really simple statement but I can't think of the answer so any help would be appreciated.