Let G be a connected, linear real reductive group. We give sufficient conditions ensuring the well-definedness of the delocalized eta invariant associated to a Dirac operator D on a cocompact G-proper manifold X and to the orbital integral 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 the orbital integral 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 associated to a cuspidal parabolic subgroup P < G with Langlands decomposition P = M A N 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.
Higher orbital integrals, rho numbers and index theory / Piazza, Paolo; Posthuma, Hessel; Song, Yanli; Tang, Xiang. - In: MATHEMATISCHE ANNALEN. - ISSN 0025-5831. - (2024). [10.1007/s00208-024-03008-2]
Higher orbital integrals, rho numbers and index theory
Piazza, Paolo;
2024
Abstract
Let G be a connected, linear real reductive group. We give sufficient conditions ensuring the well-definedness of the delocalized eta invariant associated to a Dirac operator D on a cocompact G-proper manifold X and to the orbital integral 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 the orbital integral 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 associated to a cuspidal parabolic subgroup P < G with Langlands decomposition P = M A N 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.File | Dimensione | Formato | |
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