  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.
