# An equivariant Atiyah-Patodi-Singer index theorem for proper actions I: the index formula

## TL;DR

This paper extends the Atiyah-Patodi-Singer index theorem to equivariant settings involving proper group actions on manifolds with boundary, defining a numerical index for each group element and establishing conditions for the theorem's validity.

## Contribution

It introduces an equivariant index theorem for proper group actions, generalizing classical results and providing conditions under which the theorem applies.

## Key findings

- Defined a numerical index _g(D) for equivariant elliptic operators.
- Proved the equivariant Atiyah-Patodi-Singer index theorem under specific conditions.
- Connected the equivariant index to K-theoretic indices via orbital integrals.

## 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 $g \in G$, we define a numerical index $\operatorname{index}_g(D)$, 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, $\operatorname{index}_g(D)$ can be recovered from a $K$-theoretic index of $D$ via a trace defined by the orbital integral over the conjugacy class of $g$.