  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.
