Let $K$ be a connected compact Lie group, and suppose that $(X,J,\omega)$ is a compact Kähler manifold on which the group $K$ acts holomorphically such that the group $K$ preserves the Kähler form and is Hamiltonian with momentum map $\mu: X \rightarrow \mathfrak{k}^*$.
The fixed point set $X^k:=\{x\in X|k.x=x\}$ is a Kähler manifold for any $k\in K.$
Denote $Z(k)$ by the centralizer of $k\in K$.
As a sympletic manifold, the tuple $(X^k,Z(k),\omega|_{X^k},\mu|_{X^k})$ is Hamiltonian with momentum map $\mu|_{X^k}: X^k \rightarrow {\text{Lie}(Z(k))}^*$, and we can extend the $Z(k)$-action on $X^k$ to the complexification $Z_{\mathbb{C}}(k)$.
My Question is the following:
Does the twisted product $K^{\mathbb{C}}\times_{Z(k)^{\mathbb{C}}} X^k$ have a natural Kähler structure, where $Z(k)^{\mathbb{C}}$ is the complexified Lie group of the centralizer $Z(k)$ of $k\in K$?
