Résumé
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 AdK-invariant convex function f on k & lowast;, the dual of Lie algebra of K, and study the structure of the stabilizer of the critical point off composing with the moment map. As an outcome, we are able to obtain a general Calabi-Matsushima decomposition based only on the convexity of f so that all existing Calabi-Matsushima type of decomposition theorems fall into this new framework. This work is motivated by the work of Donaldson [Don17] together with the goal of finding a natural interpretation of Tian-Zhu's Calabidecomposition for Ka<spacing diaeresis>her-Ricci solitons in [TZ02], which are examples of infinite dimensional version of our setting.