Let $\mathfrak{g}$ be a semisimple (or simple) Lie algebra over $\mathbb{C}$. Take a Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$. Let $\Phi$ be the root system of $\mathfrak{g}$ w.r.t. $\mathfrak{h}$, and $\Delta\subset\Phi$ be a set consisting of simple roots. Let $\lambda\in\Lambda_+$ be a integral dominant weight, i.e. $<\lambda,\alpha>$ are nonnegative integers for any simple root $\alpha\in\Delta$.
We know there is an unique irreducible highest weight module $L(\lambda)$ with highest wight $\lambda$.
A well-known result is:
$L(\lambda)$ is finite dimensional, and $$ \operatorname{dim}L(\lambda)_\mu=\operatorname{dim}L(\lambda)_{w(\mu)},\ \ \forall w\in W, \mu\in \mathfrak{h}^*.$$
My question is: for what kind of integral dominant weight $\lambda\in\Lambda_+$, can we say:$$ \operatorname{dim}L(\lambda)_\mu=\operatorname{dim}L(\lambda)_{\nu}\qquad \operatorname{iff}\qquad \mu\in W.\nu $$
I try to use Kostant formulas but it seems to be hard to get some useful information. I post that formula below:
$$\operatorname{dim}\ L(\lambda)_\mu=\sum_{w\in W}(-1)^{l(w)}p(\mu+\rho-w(\lambda+\rho))$$where $p=\operatorname{ch}M(0)$, the formal character of Verma module $M(0)$, and $\rho$ is one half of the sum of all positive roots.