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