Equivariant characteristic classes in the Cartan model (Q2754332)

Suppose \(G\) and \(S\) are two compact Lie groups. An \(S\)-equivariant principal \(G\)-bundle \(\pi:P\to M\) induces a principal \(G\)-bundle \(\pi_S: P_S\to M_S\) over the homotopy quotient \(M_S\). The equivariant characteristic classes of \(P\to M\) are defined to be the corresponding ordinary characteristic classes of \(P_S\to M_S\). There is also a differential geometric definition of equivariant characteristic classes in terms of the curvature of a connection on \(P\) [see \textit{N. Berline} and \textit{M. Vergne}, C. R. Acad. Sci., Paris, Sér. I 295, 539-541 (1982; Zbl 0521.57020)]. The purpose of this note is to show the compatibility of the usual differential geometric formulation of equivariant characteristic classes with the topological one above. Here the Weil model, the Cartan model, and the Mathai-Quillen isomorphism between them are used. The exposition has been tried to make as self-contained as possible in frames of a note.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00008].