The concept moment map plays a central role in the study of Hamiltonian actions of compact Lie groups $K$ on symplectic manifolds. In this note, we propose a theory of moment maps coupled with an $\Ad_K$-invariant convex function $f$ on $\fk^\ast$, the dual of Lie algebra of $K$, and study the structure of the stabilizer of the critical point of $f\circ\mu$. Our work is motivated by the work of Donaldson \cite{Donaldson2017}, which is an example of an infinite dimensional version of our setting. As an application, we give a natural interpretation of Tian-Zhu's generalized Futaki-invariant and Calabi-decomposition.
