An Equivariant Index Formula for Almost CR Manifolds

We 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 $\mathcal J(E,X)$ whose properties we describe. When E is equipped with a complex structure, we define a class of symbol mappings $\sigma$ 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 $\mathcal J(E,X)$ and standard equivariant characteristic forms. This formula generalizes the formula given in [10] for the case of a contact manifold.