Chern–Simons theory and cohomological invariants of representation varieties

We prove a general local rigidity theorem for pull-backs of homogeneous forms on reductive symmetric spaces under representations of discrete groups. One application of the theorem is that the volume of a closed manifold locally modelled on a reductive homogeneous space G/H is constant under deformation of the G/H-structure. The proof elaborates on an argument given by Labourie for closed anti-de Sitter 3-manifolds. The core of the work is a reinterpretation of old results of Cartan, Chevalley and Borel, showing that the algebra of G-invariant forms on G/H is generated by “Chern–Weil forms” and “Chern–Simons forms”.