Box splines, tensor product multiplicities and the volume function

Abstract

We study the relationship between the tensor product multiplicities of a compact semisimple Lie algebra $\\mathfrak{g}$ and a special function $\\mathcal{J}$ associated to $\\mathfrak{g}$, called the volume function, which arises in symplectic geometry and random matrix theory. Building on box spline deconvolution formulae of Dahmen-Micchelli and De Concini-Procesi-Vergne, we develop new techniques for computing the multiplicities from $\\mathcal{J}$, answering a question posed by Coquereaux and Zuber. In particular, we derive an explicit algebraic formula for a large class of Littlewood-Richardson coefficients in terms of $\\mathcal{J}$. We also give analogous results for weight multiplicities, and we show a number of further identities relating the tensor product multiplicities, the volume function and the box spline. To illustrate these ideas, we give new proofs of some known theorems.