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The McKay correspondence mentioned in Hailong's and Mike's answers extends to maximal Cohen-Macaulay modules over the invariant rings $R=k[x,y]^G$, where $G$ is a finite subgroup of $GL(2,k)$ (with $|G|$ invertible in $k$). In particular (Herzog) all such subrings have only finitely many non-isomorphic indecomposable MCM modules, that is, they have finite CM type. The converse is true in characteristic zero -- that a two-dimensional complete normal domain over $\mathbb{C}$ of finite CM type is a ring of invariants -- by a result of Auslander.

The case $G \subset SL(2,k)$ corresponds to $R$ being Gorenstein, and in particular a hypersurface -- namely the ADE hypersurfaces listed in Mike's answer. The correspondence between the irreducible representations of $G$ and the components of the exceptional fiber extends to include the irreducible MCM modules, and the McKay quiver (aka Dynkin diagram) is the same as the stable Auslander-Reiten quiver. The modules are directly connected to the irreps by Auslander, and directly connected to the components of the fiber by Gonzales-Sprinberg--Verdier and Artin--Verdier, extended to the non-Gorenstein case by Esnault and Wunram.

Most of this is in the current draft of my notes with Roger Wiegand on MCM modules, chapters 4and , 5, and 6. (Ignore the geometry in Chapter 5 -- it's riddled with errors and I'm currently rewriting it. Or you're welcome to point out errors I might not have noticed yet.) The question of what happens to the Auslander-Reiten-McKay correspondence for $G \not\subset SL(2)$ is addressed in some recent papers of Iyama and Wemyss. (You only get some of the indecomposable MCMs, the so-called special ones.)

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The McKay correspondence mentioned in Hailong's and Mike's answers extends to maximal Cohen-Macaulay modules over the invariant rings $R=k[x,y]^G$, where $G$ is a finite subgroup of $GL(2,k)$ (with $|G|$ invertible in $k$). In particular (Herzog) all such subrings have only finitely many non-isomorphic indecomposable MCM modules, that is, they have finite CM type. The converse is true in characteristic zero -- that a two-dimensional complete normal domain over $\mathbb{C}$ of finite CM type is a ring of invariants -- by a result of Auslander.

The case $G \subset SL(2,k)$ corresponds to $R$ being Gorenstein, and in particular a hypersurface -- namely the ADE hypersurfaces listed in Mike's answer. The correspondence between the irreducible representations of $G$ and the components of the exceptional fiber extends to include the irreducible MCM modules, and the McKay quiver (aka Dynkin diagram) is the same as the stable Auslander-Reiten quiver. The modules are directly connected to the irreps by Auslander, and directly connected to the components of the fiber by Gonzales-Sprinberg--Verdier and Artin--Verdier, extended to the non-Gorenstein case by Esnault and Wunram.

Most of this is in the current draft of my notes with Roger Wiegand on MCM modules, chapters 4 and 5. (Ignore the geometry in Chapter 5 -- it's riddled with errors and I'm currently rewriting it. Or you're welcome to point out errors I might not have noticed yet.) The question of what happens to the Auslander-Reiten-McKay correspondence for $G \not\subset SL(2)$ is addressed in some recent papers of Iyama and Wemyss. (You only get some of the indecomposable MCMs, the so-called special ones.)