Let's say we have a finite subgroup $\Gamma \subseteq SL(2,\mathbb{C})$ and consider the quotient variety $\mathbb{C}^2/\Gamma$, which will have one of the well-known ADE or du Val surface singularities and can be embedded into $\mathbb{C}^3$ as a hypersurface with a singular point at the origin. These singularities have crepant resolutions where the exceptional fiber is a union of $\mathbb{P}^1$s connected according to an ADE Dynkin diagram. They can be obtained by repeatedly blowing up at singular points, starting with blowing up at the origin in $\mathbb{C}^3$.

When doing this it seems that after the first blowup, the exceptional curve or curves you get correspond to the natural representation of $\Gamma$ on $\mathbb{C}^2$ that is given by its inclusion into $SL(2,\mathbb{C})$. (This is under the McKay correspondence, which gives a one-to-one correspondence between nodes in the Dynkin diagram and nontrivial irreducible representations of $\Gamma$ over $\mathbb{C}$.) For instance, if you start with an $A_n$ singularity, defined by $x^2+y^2+z^{n+1}=0$ in $\mathbb{C}^3$, and blow up at the origin, the exceptional divisor will be $X^2+Y^2=0$ in $\mathbb{P}^2$ which is two $\mathbb{P}^1$s meeting at a point. These two $\mathbb{P}^1$s correspond to the outermost nodes in the $A_n$ Dynkin diagram which are dual representations of $\Gamma$, the cyclic group of order $n+1$ in this case, that you get from its natural action on $\mathbb{C}^2$. (The node or nodes for each $\Gamma$ that I'm describing are also the node(s) that you would connect to the additional node on an extended Dynkin diagram.) Although I haven't checked recently I believe this also works for all the other ADE groups.

My question is, is there some kind of "natural" explanation for this? I was hoping this might be explained by one of the geometric interpretations of the McKay correspondence out there, like the Bridgeland-King-Reid "Mukai implies McKay" paper (which I don't thoroughly understand).