The Chern character in the simplicial de Rham complex (Q266995)

Let \(G\) be a Lie group. It is well known that the characteristic classes of principal \(G\)-bundles are in one-one correspondence with elements in \(H^*(BG)\), where \(BG\) is the classifying space of \(G\). Since it is well known that \(BG\) is not a manifold, \(H^*(BG)\) cannot be described using de Rham theory. Instead, we can consider the simplicial manifold \(NG(\ast)\) and consider a double complex \(\Omega^{p, q}(NG)=\Omega^q(NG(p))\) introduced by Dupont on it whose cohomology is isomorphic to \(H^*(BG)\). The article under review gives an explicit cocycle representative of the universal Chern character for \(G=\mathrm{GL}(n; \mathbb{C})\) in \(\Omega^{*, *}(NG)\).