Higher orbital integrals, rho numbers and index theory

2021/08/02 by Paolo Piazza, Piazza, Paolo, Hessel Posthuma +5
Mathematics · #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #K-Theory and Homology (math.KT) #Operator Algebras (math.OA) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2108.00982

openalex publication_date 2021/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 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 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^P_g associated to a cuspidal parabolic subgroup P

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