Fiber bundles associated with Anosov representations

Anosov representations $\rho$ of a hyperbolic group $\Gamma$ into a semisimple Lie group $G$ are known to admit cocompact domains of discontinuity in flag varieties $G/Q$, endowing the compact quotient manifolds $M_\rho$ with a $(G,G/Q)$-structure. In general the topology of $M_\rho$ can be quite complicated. In this article, we consider the case when $\Gamma$ is the fundamental group of a closed (real or complex) hyperbolic manifold $N$ and $\rho$ is a deformation of a (twisted) lattice embedding $\Gamma \to \mathrm{Isom}(\mathbb H_\mathbb K) \to G$ through Anosov representations. We prove that, in this situation, $M_\rho$ is alway a smooth fiber bundle over $N$. Determining the topology of the fiber seems hard in general. The second part of the paper focuses on the special case when $N$ is a surface, $\rho$ a quasi-Hitchin representation into $\mathrm{Sp}(4,\mathbb C)$, and $M_\rho$ is modelled on the space of complex Lagrangians in $\mathbb C^4$. We show that, in this case, the fiber is homeomorphic to $\mathbb{CP}^2 \sharp \overline{\mathbb{CP}^2}$.