An Euler–Maclaurin formula for the multiplicities of the equivariant index

Let $M$ be a manifold with an action of a torus $G$. If $A$ is an elliptic (or transversally elliptic) operator on $M$, invariant under $G$, the equivariant index of $A$ is a virtual representation of $G$. We express it as a sum of characters, $\mathop{\rm index}(A)(g) = \sum_{\lambda \in \hat{G}} m(\lambda) g^{\lambda}$, and obtain a function
$$ m\colon \hat{G} \to \mathbb{Z}. $$
From the Chern character of the symbol of $A$, we produce a piecewise polynomial function
$$ M\colon Lie(G)^* \to \mathbb{R}. $$
The function $M$ restricted to $\hat{G}$ coincides with $m$ (under some simplifying assumptions).
This work in progress extends some common preceding work with De Concini–Procesi.