Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$, and $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$. Let $\lambda\in\mathfrak{h}^*$ be a dominant integral weight of $\mathfrak{g}$ relative to the CSA $\mathfrak{h}$. We know there exists an unique irreducible highest weight module $L(\lambda)$ of weight $\lambda$. Since $\lambda$ is dominant integral, we know $\dim L(\lambda)<\infty$.
Define $supp(L(\lambda)):=\{\mu\in\mathfrak{h}^*\ |\ L(\lambda)_\mu\neq0\}$, and $A_i:=\{\mu\in\mathfrak{h}^*\ |\ \dim L(\lambda)_\mu=i\}$. So, we have $supp(L(\lambda))=\cup_{i=1}^\infty A_i$.
We already know that the Weyl group $W$ can act on $supp(L(\lambda))$, and each $i$ is an union of finite many $W$-orbits. My question is:
Fix a positive integer $i$. How to decompose $A_i$ into different $W$-orbits? How many $W$-orbits dose $A_i$ contain?
If it is too hard, how about A-type or simply-laced type (i.e. A/D/E-type)? How about when $i=1$?
You can describe, in some sense "tautologically", the orbits in the $A_i$ using their dominant reprentatives. (Recall: every integral weight is conjugated to a unique dominant one).
Consequently you reduced to finding multiplicities of dominant weight eigenspaces in $L(\lambda)$, that is a standard (but hard and non really explicit) question.
I think that the better answers in for general dominant weights are given by the classical Weyl, Kostant and Steinberg formulae for multplicities in $L(\lambda)$.
These formulae give a recursive way to compute multiplicities, very useful in explicit computations and special cases.