An equivariant index for proper actions I

TL;DR

This paper introduces a new equivariant index applicable to proper actions without requiring compactness of the group or orbit space, extending previous definitions and enabling broader applications.

Contribution

It develops a generalized equivariant index for proper actions that does not assume compactness, expanding the scope of index theory for deformed Dirac operators.

Findings

Generalizes equivariant index to non-compact group actions

Extends Braverman's index to broader settings

Lays groundwork for future applications in parts II and III

Abstract

Equivariant indices have previously been defined in cases where either the group or the orbit space in question is compact. In this paper, we develop an equivariant index without assuming the group or the orbit space to be compact. This allows us to generalise an index of deformed Dirac operators, defined for compact groups by Braverman. In parts II and III of this series, we will explore some properties and applications of this index.