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 $\widehat G$ of irreducible representations of $G$. Let $T$ be a maximal torus of $G$ with Lie algebra $\mathfrak t$. We construct a finite number of piecewise polynomial functions on $\mathfrak t^*$, and give a formula for the multiplicity in terms of these functions. The main new concept is the formal equivariant $\widehat A$ class.