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

作者：Hochs, Peter^*; Wang, Bai-Ling; Wang, Hang

来源：International Mathematics Research Notices, 2023, 2023(4): 3138-3193.

DOI：10.1093/imrn/rnab324

• DIRAC OPERATORS
• ETA-INVARIANTS
• SIGNATURE DEFECTS
• APS INDEX
• MANIFOLDS
• FAMILIES
• VALUES
• SPACES
• CUSPS

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 is an element of G, we define a numerical index index(g)(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, index(g)(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.