  Let G be a connected compact Lie group acting on a manifold M and let D be a
transversally elliptic operator on M. The multiplicity of the index of D is a
function on the set of irreducible representations of G. Let T be a maximal
torus of G with Lie algebra Lie(T). We construct a finite number of piecewise
polynomial functions on the dual vector space Lie(T)*, and give a formula for
the multiplicity in term of these functions. The main new concept is the formal
equivariant $\hat A$ class.
