We describe 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 are related to regular subsets of the corresponding complexified Cartan algebras.

This version fills a gap in the proof of the main results pointed out by Michael Wemyss. The only changes are footnotes on pages 3,5 and the addition of Remark 3.7