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 ${\mu }_{\mathfrak{p}}$ associated with the momentum map, we establish a Calabi-Matsushima type decomposition for the Lie algebra of the stabilizer $G_z$ at a critical point of $f\circ {\mu }_{\mathfrak{p}}$, where $f$ is the restriction to $\mathfrak{p}$ of a suitable $A{d}_U$-invariant strictly convex function on $i\mathfrak{u}$. More precisely, we show that the isotropy algebra decomposes into eigenspaces corresponding to nonnegative eigenvalues of an adjoint endomorphism, with zero eigenspace given by its reductive part. This extends the classical decomposition for the complexified action of $U^{\mathbb{C}}$ to the real reductive setting. As an application, we prove that the identity component of the compact stabilizer at critical points of $f\circ {\mu }_{\mathfrak{p}}$ is a maximal compact subgroup of the identity component of the $G$-stabilizer.