Partition functions and the index

Given a sequence X = (a1, . . . , am) of positive integers, the partition function PX(n) computes the number of solutions of the equation Xm j=1 xiai = n, xi ∈ Z ≥0 . More generally we can consider the vector case in which X consists of vectors in a lattice Λ (lying on the same side of some hyperplane). In relation to this problem, we will discuss how one is led to study some modules of functions on Λ which, among other thing, can be used to give a simple proof of the local quasi-polynomiality of PX. Following some joint work with Procesi and Vergne, if G is the torus having Λ as character group, M the linear representation of G, whose list of weights is X, these modules give the range of the index map for Gtransversally elliptic operators on M.