A Calabi--Matsushima decomposition for real reductive group actions

Let $Z$ be a compact Kähler manifold endowed with a Hamiltonian action of a compact connected Lie group $U$, and let $G\subset U^{\mathbb C}$ be a real compatible subgroup. Using the gradient map $μ_{\mathfrak p}$ associated with the momentum map, we establish a Calabi-Matsushima type decomposition