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 $\eta_g (D_X)$ associated to a Dirac operator $D_X$ on a cocompact $G$-proper manifold $X$ and to the orbital integral $\tau_g$ defined by a semisimple element $g\in G$. Along the way, we give a detailed account of the large time behaviour of the heat kernel and of its short time bahaviour 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 $\tau_g$ 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 $\Phi^P_g$ associated to a cuspidal parabolic subgroup $P<G$ with Langlands decomposition $P=MAN$ and a semisimple element $g\in M$. We employ these results in order to define (higher) rho numbers associated to $G$-invariant positive scalar curvature metrics.