Let $X$ be a Riemannian manifold and let $G$ be a (at most countable, if that matters) discrete group acting properly and by isometries on $X$. Let $\mathcal{O}$ be the sheaf of analytic functions on $X$. By analogy with what happens for finite groups acting on vector spaces, one is tempted to study a sheaf written $\mathcal{O} \rtimes G$. What is a good reference for an algebraist to learn about the various convergence conditions one might impose to define this sheaf, and their relationship with the quotient X/G?

[I'm fine with an answer that works in a different category---e.g. complex analytic spaces, but I want there to be some convergence conditions imposed at some point. The ideal reference is a short survey paper with precise definitions.]