Let \(\mathcal {G}\) be a fusion category acting on a triangulated category \(\mathcal {D}\) , in the sense that \(\mathcal {D}\) is a \(\mathcal {G}\) -module category. Our motivation example is fusion-weighted species, which is essentially Heng’s construction. We study \(\mathcal {G}\) -stable tilting, cluster and stability structures on \(\mathcal {D}\) . In particular, we prove the deformation theorem for \(\mathcal {G}\) -stable stability conditions. A first application is that Duffield–Tumarkin’s categorification of cluster exchange graphs of finite Coxeter–Dynkin type can be naturally realized as fusion-stable cluster exchange graphs. Another application is that the universal cover of the hyperplane arrangements of any finite Coxeter–Dynkin type can be realized as the space of fusion-stable stability conditions for certain ADE Dynkin quiver. This provides an alternative uniform proof of \(K(\pi ,1)\) -conjecture in the finite Coxeter–Dynkin case.
