I shall explain how cyclic cohomology and K-theory can be used in order to define and investigate primary and secondary invariants of G-invariant Dirac operators on a cocompact G-proper manifold, with G a connected real reductive Lie group . After treating the case of cyclic cocycles associated to elements in the differentiable cohomology of G I will move to delocalized cyclic cocycles; in particular, I will explain the challenges in defining the delocalized eta invariant associated to the orbital integral defined by a semisimple element g in G and in showing that such an invariant enters in an Atiyah-Patodi-Singer index theorem for cocompact G-proper manifolds. I will then move to a higher version of these results, using the cyclic cocycles of Song and Tang associated to the higher orbital integral defined by the choice of a cuspidal parabolic subgroup P<G with Langlands decomposition P=MAN and a semisimple element g in M.
This talk is based on articles with Hessel Posthuma and with Hessel Posthuma, Yanli Song and Xiang Tang.