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Let $\mathfrak{g}$ a finite-dimensional complex simple Lie algebra with Cartan subalgebra $\frak h$. Let denote $(\cdot,\cdot)$ the non-degenerate bilinear form on $\frak h^*$ induced by the Killing form of $\frak g$. Denote by $R$, $P$ and $P^+$ the set of roots, the weight lattice and the set of dominant weights of $\mathfrak{g}$, respectively.

Given $\mu \in P^+$, define $$P(\mu)= \{\alpha \in R \mid (\alpha,\mu) = \min_{\beta\in R}(\beta,\mu)\}.$$

Is there a constructive way to describe this set $P(\mu)$? Where can I find something related?

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