In a problem I am trying to solve, the following situation occurs. $X$ is a smooth variety and $G$ is a reductive group acting transitively on $X$. We have the stack $X/G$ and a morphism $\pi : X \to X / G$. Fix a basepoint $\* \in X$ and let $H$ be the stabilizer of $\*$. I don't know much about stacks but I was wondering if the following statements are correct and if someone could give me a reference where I can read about them:


* The quasi-coherent sheaves on $X / G$ are the same as $H$-representations. Here I am thinking of the fact that an equivariant vector bundle on a homogeneous manifold is determined by its fiber over some chosen basepoint. I guess here we could also talk about the action groupoid, but since the action is transitive i don't think that is needed
* Let $K$ be the fraction field of $X$ and $V$ an $H$-representation. The rational sections of $\pi^\* V$ are $ (K \otimes\_{\mathbb{C}} V)^H$. I must admit, this is not geometrically obvious to me even in the case of manifolds, but it is all I could think of writing down. I also asked some people in my department and they seemed to think that this was reasonable.
* If $X$ is affine with coordinate ring $A$, then the global sections of $\pi^\*V$ are $(A \otimes\_{\mathbb{C}} V)^H$. Again, this is not obvious to me, but it seems like the obvious thing to write down.


All of this is probably easy if you know about stacks, so hopefully someone will share some of their stacky wisdom!



Pulling back quasi-coherent sheaves from a quotient stack
For the first question, there is an equivalence of stacks between $X/G$ and $\*/H$, and Theorem 4.46 of [Vistoli's notes](http://arxiv.org/abs/math/0412512) gives an equivalence between $H$-equivariant quasicoherent sheaves on a point (i.e., representations of $H$) and quasicoherent sheaves on $\*/H$.


Your conjectures about rational and global sections seem fail when $X$ is a point. In particular, pullback from $\*/H$ to a point $\*$ is what gives an equivalence between quasicoherent sheaves on $\*/H$ and $H$-equivariant sheaves on $\*$, and taking fixed points will exclude all non-trivial $H$-modules from your category. In general, you get the $G$-modules $K \otimes\_{\mathbb{C}} V = K \otimes\_A \operatorname{Ind}\_{H}^{G} V$ and $A \otimes\_{\mathbb{C}} V = \operatorname{Ind}\_{H}^{G} V$, because tensoring an $H$-module $V$ with $A$ over $\mathbb{C}$ is the same as tensoring the $\mathcal{O}(H)$-comodule $V$ with $\mathcal{O}(G)$ over $\mathcal{O}(H)$.


