A Hitchin-Kobayashi correspondence for Kaehler fibrations

Let $X$ be a compact Kaehler manifold and $E\to X$ a principal $K$ bundle, where $K$ is a compact connected Lie group. Let ${\cal A}^{1,1}$ be the set of connections on $E$ whose curvature lies in $\Omega^{1,1}(E\times_{Ad} {\frak k})$, where ${\frak k}$ is the Lie algebra of $K$. Endow $\frak k$ with a nondegenerate biinvariant bilinear pairing. This allows to identify \{\frak k}\simeq{\frak k}^*. Let $F$ be a Kaehler left $K$-manifold and suppose that there exists a moment map $\mu$ for the action of $K$ on $F$. Let ${\cal S}=\Gamma(E\times_K F)$. In this paper we study the equation $$\Lambda F_A+\mu(\Phi)=c$$ for $A\in {\cal A}^{1,1}$ and a section $\Phi\in {\cal S}$, where $c\in{\frak k}$ is a fixed central element. We study which orbits of the action of the complex gauge group on ${cal A}^{1,1}\times{\cal S}$ contain solutions of the equation, and we define a positive functional on ${cal A}^{1,1}\times{\cal S}$ which generalises the Yang-Mills-Higgs functional and whose local minima coincide with the solutions of the equation.