Stability Conditions and Kleinian Singularities

We describe (connected components of) the spaces of stability conditions on certain triangulated categories associated to Dynkin diagrams. These categories can be defined either algebraically via module categories of preprojective algebras, or geometrically via coherent sheaves on resolutions of Kleinian singularities. The resulting spaces of stability conditions are covering spaces of regular subsets of the corresponding complexified Cartan algebras.