The eta invariant and equivariant index of transversally elliptic operators

Abstract

We prove 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, we obtain an index formula for basic Dirac operators on Riemannian foliations, a problem that was open for many years.

arXiv comment62 pages, typos corrected

References in corpus (3)

• Hopf algebras, cyclic cohomology and the transverse index theory
• Desingularizing compact Lie group actions
• Equivariant cohomology and resolution

Cited by in corpus (7)

• Modified differentials and basic cohomology for Riemannian foliations
• Index theory for basic Dirac operators on Riemannian foliations
• The equivariant index theorem for transversally elliptic operators and the basic index theorem for Riemannian foliations
• Desingularizing compact Lie group actions
• The eta invariant on two-step nilmanifolds
• Transversal Dirac operators on distributions, foliations, and G-manifolds: Lecture notes
• A Local Index Theorem of Transversal Type on Manifolds with Locally Free ${\mathbb{S}}^1$-action