I am trying to understand the basics of equivariant cohomology in view of applications to the field of crystalline topological insulators. At stake in that field is the very explicit situation of the bundle of eigenvalues of a hermitian matrix depending upon parameters belonging to a compact space and invariant by some symmetry group.
The good tool seems to be equivariant cohomology and in particular equivariant Chern classes. In the physics literature several formulas definitely look like localization formulas.
There are several textbooks on equivariant cohomology (e.g. the recent book by Loring Tu) where there is an equivariant de Rahm theorem that is discussed.
My question: these results involve equivariant cohomology with trivial (i.e. non twisted) coefficients. Now there is a paper by B. Kahn ('Construction de classes de Chern equivariantes pour un fibré vectoriel réel') where the Picard group of equivariant line bundles is shown to be isomorphic to the equivariant cohomology group of order 2 with twisted coefficients Z(1). This paper seems to be scarcely cited in the literature devoted to equivariant cohomology. However in the paper "Conjugation spaces and equivariant Chern classes" by W. Pitsch and J. Scherer it is shown that the coefficients should be twisted in the case of Real bundles in the sense of Atyiah. Could someone give me a broader view and explain how both approaches might fit in the same (larger?) framework?
Subsidiary question: I am not sure I understand what Z(j) is as a local coefficients system, in particular what exactly is Z(0)
Any help would be much appreciated.