Abstract : I shall explain how 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. This involves cyclic cohomology and Ktheory. 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 consider a higher version of these results, based on the Song-Tang higher orbital integrals associated to 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 Postrhuma, Yanli Song and Xiang Tang.

Keywords : Lie groups; proper actions; k-theory; cyclic cohomology; Dirac operators