  Let G be a connected, linear real reductive group and let X be a G-proper
manifold without boundary. We give a detailed account of both the large and
small time behaviour of the heat-kernel of perturbed Dirac operators, as a map
from the positive real line to the Lafforgue algebra. As a first application,
we prove that certain delocalized eta invariants associated to perturbed Dirac
operators are well defined. We then prove index formulas relating these
delocalized eta invariants to Atiyah-Patodi-Singer delocalized indices on
G-proper manifolds with boundary. We apply these results to the definition of
rho-numbers associated to G-homotopy equivalences between closed G-proper
manifolds and to the study of their bordism properties.
