On statistics of irreducible components in large tensor products of modules over simple Lie algebras

Let $V_1,\dots, V_k$ be finite dimensional $\mathfrak g$-modules where $\mathfrak g$ is a finite dimensional Lie algebra and $N_1,\dots, N_k$ are positive integers. In the talk the asymptotical behavior of multiplicities of irreducible representations will be computed in the limit $N_i\to \infty$ with $N_1:\dots: N_k$ being finite. As a consequence the asymptotical probability distribution of irreducible representations with respect to the Plancherel and other natural probability measures will be established.