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

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 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.