The Littlewood-Richardson coefficients describe the decomposition of tensor products of irreducible representations of a simple Lie algebra into irreducibles. Assuming the number of factors is large, one gets a measure on the space of weights. This limiting measure was extensively studied by many authors. In particular, Kerov computed the corresponding density in a special case in type A and Kuperberg gave a formula for the general case. The goal of this paper is to give a short, self-contained and pure Lie theoretic proof of the formula for the density of the limiting measure. Our approach is based on the link between the limiting measure induced by the Littlewood-Richardson coefficients and the measure defined by the weight multiplicities of the tensor products.
Let $G$ be a connected reductive group acting on an irreducible normal algebraic variety $X$. We give a slightly improved version of Local Structure Theorems obtained by Knop and Timashev, which describe the action of some parabolic subgroup of $G$ on an open subset of $X$. We also extend various results of Vinberg and Timashev on the set of horospheres in $X$. We construct a family of nongeneric horospheres in $X$ and a variety $\Hor$ parameterizing this family, such that there is a rational $G$-equivariant symplectic covering of cotangent vector bundles $T^*\Hor \dashrightarrow T^*X$. As an application we recover the description of the image of the moment map of $T^*X$ obtained by Knop. In our proofs we use only geometric methods which do not involve differential operators.
The unitary group U(N) acts by conjugations on the space H(N) of NxN Hermitian matrices, and every orbit of this action carries a unique invariant probability measure called an orbital measure. Consider the projection of the space H(N) onto the real line assigning to an Hermitian matrix its (1,1)-entry. Under this projection, the density of the pushforward of a generic orbital measure is a spline function with N knots. This fact was pointed out by Andrei Okounkov in 1996, and the goal of the paper is to propose a multidimensional generalization. Namely, it turns out that if instead of the (1,1)-entry we cut out the upper left matrix corner of arbitrary size KxK, where K=2,...,N-1, then the pushforward of a generic orbital measure is still computable: its density is given by a KxK determinant composed from one-dimensional splines. The result can also be reformulated in terms of projections of the Gelfand--Tsetlin polytopes.