This is a question related to the two dimensional McKay correspondence. Let $R = \mathbb{C}[x,y]$, and $G$ a finite group acting on $R$. Recall that a $G$-equivariant $R$-module is an $R$-module with a $G$ action, such that the multiplication map $R \otimes M \to M$ is $G$-equivariant, i.e., $g *(rm) = (g\cdot r) (g *m)$.
It is well known that the category of such modules, $R\text{-Mod}_G$, is equivalent to the category of modules over the (noncommutative, unital) skew group algebra $R \# G$, which is the algebra on the underlying vector space $R \otimes_{\mathbb{C}} \mathbb{C}[G]$, and multiplication defined by $(r\otimes g)(r' \otimes g') = r(g\cdot r')\otimes gg'$.
I would like to know what the projective objects are in the latter category. I know that Quillen-Suslin implies that finitely generated projective $R$-modules are free, and so I suspect that all the finitely generated projective equivariant $R$-modules are also free. However I would like to know a proof of this in the setting of the skew group ring. I am aware that the group ring $\mathbb{C}[G]$ is semisimple, so one would hope that $R \# G$ also has a similar decomposition, but I can only see this as vector spaces, rather then algebras. Moreover I'm still not sure how I would use this to get projective objects in $R\# G$-Mod.