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We study the relationship between the tensor product multiplicities of acompact semisimple Lie algebra $\mathfrak{g}$ and a special function$\mathcal{J}$ associated to $\mathfrak{g}$, called the volume function. Thevolume function arises in connection with the randomized Horn's problem inrandom matrix theory and has a related significance in symplectic geometry.Building on box spline deconvolution formulae of Dahmen-Micchelli and DeConcini-Procesi-Vergne, we develop new techniques for computing themultiplicities from $\mathcal{J}$, answering a question posed by Coquereaux andZuber. In particular, we derive an explicit algebraic formula for a large classof Littlewood-Richardson coefficients in terms of $\mathcal{J}$. We also giveanalogous results for weight multiplicities, and we show a number of furtheridentities relating the tensor product multiplicities, the volume function andthe box spline. To illustrate these ideas, we give new proofs of some knowntheorems.

Box splines, tensor product multiplicities and the volume function