Higher orbital integrals, rho numbers and index theory

Abstract

Let G be a connected, linear, reductive Lie group. We give sufficient conditions ensuring the well-definedness of the delocalized eta invariant ηg(D X ) associated to a Dirac operator D X on a cocompact G-proper manifold X and to the orbital integral τg defined by a semisimple element g ∈ G. We prove that such an invariant enters in the the boundary correction term in a number of index theorems computing the pairing between the index class and the 0-degree cyclic cocycle defined by τg on a G-proper manifold with boundary. We 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 P g associated to a cuspidal parabolic subgroup P < G with Langlands decomposition P = M AN 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 and to G-equivariant homotopy equivalences. Contents1. Introduction 2. Geometric preliminaries. 3. Delocalized traces and the APS index formula 3.1. Orbital integrals and associated cyclic 0-cocycles in the closed case 3.2. Manifolds with boundary: smooth index classes 3.3. 0-degree (relative) cyclic cocycles associated to orbital integrals 3.4. The 0-degree delocalized APS index theorem on G-proper manifolds 4. Delocalized eta invariants for G-proper manifolds 4.1. Large time behaviour 4.2. Small time behavior 4.3. Short time limits 5. Modified delocalized eta invariants 5.1. Spectrum on the slice 5.2. Delocalized eta invariants associated to perturbations on the slice 5.3. Delocalized eta invariants under a gap assumption 5.4. Delocalized eta invariants associated to smoothing perturbations 6. Higher delocalized cyclic cocycles 7. Toward a general higher APS index formula 8. Reduction 9. An index theorem for higher orbital integral through reduction 10. The non-invertible case 10.1. The gap case for 0-degree cyclic cocycles associated to orbital integrals 10.2. The gap case for cyclic cocycles associated to higher orbital integrals 10.3. Perturbations from the slice when G is equal rank 10.4. Perturbations from the slice for general G 11. Numeric rho invariants on G-proper manifolds 11.1. Rho numbers associated to delocalized 0-cocycles