An equivariant index formula for almost-CR manifolds

TL;DR

This paper develops an equivariant index formula for almost-CR manifolds with a compact Lie group action, generalizing previous contact manifold results by incorporating transversality and complex structures.

Contribution

It introduces a new index formula for H-transversally elliptic symbols on almost-CR manifolds, extending prior contact manifold formulas to broader geometric settings.

Findings

Derived a formula involving J(E,X) and characteristic classes for the equivariant index.

Generalized previous contact manifold index formulas to almost-CR structures.

Provided properties of the equivariant differential form J(E,X) in this context.

Abstract

We consider a consider the case of a compact manifold M, together with the following data: the action of a compact Lie group H and a smooth H-invariant distribution E, such that the H-orbits are transverse to E. These data determine a natural equivariant differential form with generalized coefficients J(E,X) whose properties we describe. When E is equipped with a complex structure, we define a class of symbol mappings in terms of the resulting almost-CR structure that are H-transversally elliptic whenever the action of H is transverse to E. We determine a formula for the H-equivariant index of such symbols that involves only J(E,X) and standard equivariant characteristic classes. This formula generalizes the formula given in arXiv:0712.2431 for the case of a contact manifold.