Higher orbital integrals, rho numbers and index theory

Let G be a connected, linear real reductive group. We give sufficient conditions ensuring the well-definedness of the delocalized eta invariant η_g(D_X) associated to a Dirac operator D_X on a cocompact G-proper manifold X and to the orbital integral τ_g defined by a semisimple element g ∈ G. Along the way, we give a detailed account of the large time behaviour of the heat kernel and of its short time behaviour near the fixed point set of g. We prove that such a delocalized eta invariant enters as the boundary correction term in an index theorem computing the pairing between the index class and the 0-degree cyclic cocycle defined by τ_g on a G-proper manifold with boundary. More importantly, we also prove a higher version of such a theorem, for the pairing of the index class and the higher cyclic cocycles defined by the higher orbital integral Φ_g^P associated to a cuspidal parabolic subgroup P < G with Langlands decomposition P = MAN and a semisimple element g ∈ M. We employ these results in order to define (higher) rho numbers associated to G-invariant positive scalar curvature metrics.