Abstract: For both elliptic and transversally elliptic operators, we have seen that the equivariant index defines a virtual $G$-representation, where $G$ is a compact Lie group. This representation can be expressed as a sum of irreducible representations with multiplicities. In the elliptic case, this sum is finite. In the transversally elliptic case, the sum is infinite, but the index still defines a distributional character on $G$.

The aim of this talk is to give an overview of how the de Concini-Procesi-Vergne machinery (associated to the infinitesimal index) can be used to give a formula for the multiplicities of the irreducible representations within the equivariant index. This will be more of a ``big picture'' talk that attempts to tie together some of the particular results we've encountered so far, without getting too much into the details. The main reference will be Vergne's paper on the Euler-Maclaurin formula for the multiplicity function (arXiv:1211.5547).

Abstract: I will describe a project with June Huh on Chern-Schwarz-MacPherson classes of some varieties associated with matroids, and combinatorial inequalities that result.