# modules – Decompose weights of same multiplicity into different Weyl group orbits

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).

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)?