We consider the action of a compact Lie group G on (M, E), where M is a compact manifold and E is a smooth subbundle of TM. If the orbits of the group action are transverse to E, then it is possible to define an equivariant differential form with generalized coefficients that depends only on E and the group action. If M is equipped with an almost-CR structure whose underlying real subbundle is E, then we can also construct a G-transversally elliptic differential operator and compute its equivariant index. We will explain how our differential form is defined, and what it has to do with the equivariant index of this operator. Finally, provided that time has moved sufficiently slowly, we will mention some applications of this work, in particular to the quantization of contact manifolds, and the character of induced representations.