Formal equivariant $\widehat{A}$ class, splines and multiplicities of the index of transversally elliptic operators

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 $\widehat{A}$ class.