On multiplicities of irreducibles in large tensor product of representations of simple Lie algebras

In this paper, we study the asymptotics of multiplicities of irreducible representations in large tensor products of finite dimensional representations of simple Lie algebras and their statistics with respect to Plancherel and character probability measures. We derive the asymptotic distribution of irreducible components for the Plancherel measure, generalizing results of Biane and Tate and Zelditch. We also derive the asymptotics of the character measure for generic parameters and an intermediate scaling in the vicinity of the Plancherel measure. It is interesting that the asymptotic measure is universal and after suitable renormalization does not depend on which representations were multiplied but depends significantly on the degeneracy of the parameter in the character distribution.