Let $\mathfrak g$ be a semisimple Lie algebra and $\mathfrak g = \mathfrak n \oplus \mathfrak h \oplus \mathfrak n^-$ be the triangular decomposition. A weight of a finite dimensional irreducible representation of $\mathfrak g$ is called extremal if it is in the Weyl group orbit of the highest weight.

For a fundamental weight $\omega_i$, write $n_i$ for the number of extremal weights in the finite dimensional irreducible $\mathfrak g$-representation $M(\omega_i)$ corresponding to $\omega_i$. Am I correct that $\sum_i n_i$ equals the number of positive roots in the root system for $(\mathfrak g, \mathfrak h)$? If the answer is yes, how does one prove it, and is there a natural bijection between $\bigcup_i \{\text{extremal weights for } M(\omega_i)\}$ and the set of positive roots?

Thank you very much in advance!