Non-abelian Hodge correspondence and moduli spaces of flat bundles on Sasakian manifolds with fixed basic structures

We show that the moduli space of simple flat bundles over a compact Sasakian manifold is a finite disjoint union of moduli spaces of simple flat bundles with fixed basic structures. This gives a detailed description of the non-abelian Hodge correspondence on a compact Sasakian manifold at the level of moduli spaces. As an application, we give an analogue of Hitchin's properness of maps defined by the coefficients of the characteristic polynomial of Higgs fields.