Let $mathfrak{g}$ be a finite-dimensional simple Lie algebra over $mathbb{C}$, and $mathfrak{h}$ is a Cartan subalgebra of $mathfrak{g}$. Let $lambdainmathfrak{h}^*$ 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 $supp(L(lambda)):={muinmathfrak{h}^* | L(lambda)_muneq0}$, and $A_i:={muinmathfrak{h}^* | dim L(lambda)_mu=i}$. 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 (A/D/E-type)?