I'm trying to learn about stacks, but I'm sure I'm misunderstanding something about sheaves on them.
Let $\mathcal X$ be a DM stack over $\mathbb C$, and let $F$ be a sheaf of $\mathcal{O}_{\mathcal X}$-modules.
Now let $x\colon \operatorname{Spec} \mathbb C\to \mathcal X$ be a geometric point with nontrivial stabilizer (let's call the stabilizer $G_x$).
How do you understand the action of $G_x$ on the fiber of $F$ over $x$?
I'd be very happy to see a simple example like $\mathcal X=[\mathbb{C}^2/\mathbb{Z_3}]$, with $x$ as the unique point with stabilizer $G_x=\mathbb Z_3$. Let's take $F$ to be the sheaf on $\mathcal X$ descending from $\mathcal O_{\mathbb C^2}\otimes V$, with $V$ a nontrivial $\mathbb Z_3$-representation.
It seems to me that the fiber of $F$ over $x$ should just be isomorphic to $V$, but I'm really not sure how to show this properly.