Let $R$ be a commutative noetherian domain that is also an algebra over a field $k$ Let $G$ is a finite group that acts on $R$ in a non-trivial way. Let $A=R*G$ be the skew group algebra of this action.
How can one compute the trace group (aka zeroth Hochschild homology group) $$ \operatorname{tr}(A)=A/[A,A] $$ of $A$?
The special case I'm mostly interested in is the following. Fix $k=\mathbb{C}$. Let $G \subset SL(V)$ be a finite subgroup and assume that $V$ is an irreducible complex representation of $G$. Let $R=S[V]$, where action of $G$ naturally extends to the symmetric algebra $S[V]$.
Your special case is done in detail in
You can check by hand that $A/[A,A]$ is the isomorphic to $(A/[R,A])_G$; here $A/[R,A]$ is a $G$-module in an obvious way, and $M_G$ denotes $M/(gm-m:g\in G,m\in M)$ for a $G$-module. This allows you do compute what you want in two steps: first look at $A/[R,A]$, and then take the coinvariants $(-)_G$ of the result.