  We extend the Donaldson-Corlette-Hitchin-Simpson correspondence between Higgs
bundles and flat connections on compact K\"ahler manifolds to compact
quasi-regular Sasakian manifolds. A particular consequence is the translation
of restrictions on K\"ahler groups proved using the
Donaldson-Corlette-Hitchin-Simpson correspondence to fundamental groups of
compact Sasakian manifolds.

Differential Geometry;Algebraic Geometry;Complex Variables