Moment map, convex function and extremal point

The moment map $\mu$ is a central concept in the study of Hamiltonian actions of compact Lie groups $K$ on symplectic manifolds. In this short note, we propose a theory of moment maps coupled with an ${\mathrm{A}\mathrm{d}}_K$-invariant convex function $f$ on ${\mathfrak{k}}^{\ast }$, the dual of Lie algebra of $K$, and study the properties of the critical point of $f\circ \mu$. Our motivation comes from Donaldson \cite{Donaldson2017} which is an example of infinite dimensional version of our setting. As an application, we interpret Kähler-Ricci solitons as a special case of the generalized extremal metric.