The saturation property for refined Littlewood-Richardson coefficients

Given dominant integral weights $\lambda, \mu, \nu$ of a finite-dimensional simple Lie algebra $\mathfrak{g}$ and an element $w$ of its Weyl group, the refined tensor product multiplicity $c_{\lambda \mu}^\nu(w)$ is the multiplicity of the irreducible $\mathfrak{g}$-module $V(\nu)$ in the so-called Kostant--Kumar submodule $K(\lambda, w, \mu)$ of the tensor product $V(\lambda) \otimes V(\mu)$. We derive properties of these coefficients in general type, including a Brauer--Klimyk type formula and restriction theorems. In type $A$, we obtain a hive model for the $c_{\lambda \mu}^\nu(w)$ and prove that the saturation and strong semigroup properties hold if the permutation $w$ is $312$-avoiding, $231$-avoiding, or a commuting product of such elements. This generalizes the classical Knutson--Tao saturation theorem.