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We apply equivariant localisation to the theory of $Z$-stability and$Z$-critical metrics on a K\"ahler manifold $(X,\alpha)$, where $\alpha$ is aK\"ahler class. We show that the invariants used to determine $Z$-stability ofthe manifold, which are integrals over test configurations, can be written as aproduct of equivariant classes, hence equivariant localisation can be applied.We also study the existence of $Z$-critical K\"ahler metrics in $\alpha$, whoseexistence is conjectured to be equivalent to $Z$-stability of $(X,\alpha)$. Inparticular, we study a class of invariants that give an obstruction to theexistence of such metrics. Then we show that these invariants can also bewritten as a product of equivariant classes. From this we give a new, moredirect proof of an existing result: the former invariants determining$Z$-stability on a test configuration are equal to the latter invariantsrelated to the existence of $Z$-critical metrics on the central fibre of thetest configuration. This provides a new approach from which to derive the$Z$-critical equation.

Equivariant localisation in the theory of $Z$-stability for K\"ahler manifolds