Characteristic forms for holomorphic families of local systems

Let Γ \Gamma be a finitely generated group and G G a complex Lie group with Lie algebra g \mathfrak {g} . The complex variety of group homomorphisms H o m ( Γ , G ) Hom(\Gamma ,\,G) is equipped with an action of G G by inner automorphism and thus it defines a complex-analytic stack. We associate to each Φ ∈ ( S y m n g ) G \Phi \,\in \, (Sym^{n} \mathfrak {g})^{G} a closed holomorphic n n -form on this stack with values in H n ( Γ , C ) H^{n}(\Gamma ,\,\mathbb {C}) . This construction is multiplicative so that we get a graded algebra homomorphism. We obtain this as a universal refinement of a Chern-Weil homomorphism. If we take n = 2 n \,=\, 2 and let Γ \Gamma be the fundamental group of a closed orientable surface, then we recover a classical construction due to Goldman.