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

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