TY - JOUR
T1 - An Equivariant Atiyah–Patodi–Singer Index Theorem for Proper Actions I
T2 - The Index Formula
AU - Hochs, Peter
AU - Wang, Bai Ling
AU - Wang, Hang
N1 - Publisher Copyright:
© The Author(s) 2021. Published by Oxford University Press.
PY - 2023/2/1
Y1 - 2023/2/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85150024842&partnerID=8YFLogxK
U2 - 10.1093/imrn/rnab324
DO - 10.1093/imrn/rnab324
M3 - Article
SN - 1073-7928
VL - 2023
SP - 3138
EP - 3193
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
IS - 4
ER -