Say for simplicity that a variety $X$ is the quotient of $\mathbb{A}^n$ by a finite group action $G$ whose fixed points live in codimension at least $2$. Let $X^0$ be the smooth locus of $X$ and $i$ its inclusion map. I am trying to prove that the canonical sheaf $\omega_X = i_{\ast} \Lambda^n \Omega_{X^0/k}$ is given by the $G$-invariant elements of $j_{\ast}\Lambda^n \Omega_{\mathbb{A}^n/k}$, where $j$ is the inclusion of the non-fixed points into $\mathbb{A}^n$. This is an attempt to reconcile for myself the "differential" notion of the canonical sheaf of an orbifold ($G$-invariant canonical forms) with the algebraic definition of the canonical sheaf.
In fact I think this is easily done when $n = 1$: $X$ is the spectrum of the $G$-invariant regular functions on $\mathbb{A}^n$, which are easily seen to correspond to the $G$-invariant Kahler differentials. The issue is taking the tensor power. For if a group $H$ acts on a module $M$, this action extends naturally to $\Lambda^n M$, but it is not generally the case that $(\Lambda^n M)_H = \Lambda^n (M_H)$.
Any insight would be appreciated. If I misunderstand the definition of the canonical sheaf of an orbifold, that would be good to know, too. Thanks.