The equivariant index theorem for transversally elliptic operators and the basic index theorem for Riemannian foliations

TL;DR

This paper presents a formula for the index multiplicities of equivariant transversally elliptic operators on G-manifolds, involving integrals over stratified blowups and eta invariants, with applications to Riemannian foliations.

Contribution

It provides a new index formula for basic Dirac operators on Riemannian foliations, solving a long-standing open problem.

Findings

Derived a formula involving integrals over stratified blowups

Connected eta invariants to index multiplicities

Applied results to Riemannian foliations

Abstract

In this expository paper, we explain a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a $G$-manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of associated elliptic operators. Among the applications is an index formula for basic Dirac operators on Riemannian foliations, a problem that was open for many years. This paper summarizes the work in the papers arXiv:1005.3845 [math.DG] and arXiv:1008.1757 [math.DG].