I am interested in special dominant integral weights $\lambda \in \mathfrak{h}^*$, where $\mathfrak{h}$ is a Cartan subalgebra of the Lie algebra $\mathfrak{g}$ of a compact semisimple Lie group $G$.
The condition that such special dominant integral weights satisfy is:
$$ | W. \lambda | = 1 + \sum_{\alpha \in \Delta^+} h_{\alpha}(\lambda). $$
where $W$ is the Weyl group of $G$, $W.\lambda$ denotes the Weyl orbit of $\lambda$, $\Delta^+$ is the set of positive roots of $\mathfrak{g}$, and $h_{\alpha} \in \mathfrak{h}$ is the co-root associated to the root $\alpha \in \mathfrak{h}^*$.
This can be rephrased as such. Let $H$ denote the sum of positive coroots. Then the condition is:
$$ | W. \lambda | = 1 + H(\lambda). $$
What is known about such integral weights? Given a compact semisimple Lie group, does there always exist (a Weyl orbit of) such a special integral weight? How many Weyl orbits of such special integral weights exist, if any? Do they have a name? What is known about them?
Of course, I can go through the exercise, and check case-by-case. I know there exist such special integral weights for $SU(n)$ and $Sp(m)$. I did not check yet for other cases. However, given that it is more and more difficult to find free time nowadays, I figure it would be wiser to ask, rather than "reinvent the wheel", so to speak.
Edit: I realize that I want $H$ to be the sum of the positive coroots, and not half that sum. Thus $H = 2 \rho^\vee$, where $\rho^\vee$ denotes half the sum of the positive coroots. I have edited my post above accordingly.