A higher Lefschetz formula for flat bundles

In this paper, we prove a fixed point formula for flat bundles. To this end, we use cyclic cocycles which are constructed out of closed invariant currents. We show that such cyclic cocycles are equivariant with respect to isometric longitudinal actions of compact Lie groups. This enables us to prove fixed point formulae in the cyclic homology of the smooth convolution algebra of the foliation.