An Equivariant Atiyah–Patodi–Singer Index Theorem for Proper Actions I: The Index Formula

Peter Hochs*, Bai Ling Wang, Hang Wang

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    5 Citations (Scopus)

    Abstract

    Consider a proper, isometric action by a unimodular locally compact group G on a Riemannian manifold M with boundary, such that M/G is compact. For an equivariant, elliptic operator D on M, and an element g ∈ G, we define a numerical index indexg(D), in terms of a parametrix for D and a trace associated to g. We prove an equivariant Atiyah–Patodi–Singer index theorem for this index. We first state general analytic conditions under which this theorem holds, and then show that these conditions are satisfied if g = e is the identity element; if G is a finitely generated, discrete group, and the conjugacy class of g has polynomial growth; and if G is a connected, linear, real semisimple Lie group, and g is a semisimple element. In the classical case, where M is compact and G is trivial, our arguments reduce to a relatively short and simple proof of the original Atiyah–Patodi–Singer index theorem. In part II of this series, we prove that, under certain conditions, indexg(D) can be recovered from a K-theoretic index of D via a trace defined by the orbital integral over the conjugacy class of g.

    Original languageEnglish
    Pages (from-to)3138-3193
    Number of pages56
    JournalInternational Mathematics Research Notices
    Volume2023
    Issue number4
    DOIs
    Publication statusPublished - 1 Feb 2023

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