Let $Q$ be a finite quiver. As far as I know there’s a great amount of work concerning the so-called quiver varieties one can associate to it. Loosely speaking, these are obtained by taking GIT quotients of semistable representations of the (deformed) preprojective algebra one get from $Q$.
These varieties turn out to have a wide range of applications to representation theory and algebraic geometry.
Is there any work going on pursuing the “stacky” approach? Like taking already the stack of semistable representations or maybe even the full stack of representations of the preprojective algebra?