Extremal Weyl-orbit weights

Given: The Axiom of Choice, such $\mathfrak{g},\mathfrak{h}$, a fixed positive system with Weyl group $W$ and fundamental chamber $C$, a dominant integral $\lambda$, and the module $V(\lambda )$.

The Axiom of Choice is assumed; it enters through the root-space theory used by the cited suppliers (The Axiom of Choice).

$V(\lambda )$ is a finite-dimensional irreducible highest weight module of highest weight $\lambda$; its $\lambda$-weight space is one-dimensional and every weight $\mu$ of $V(\lambda )$ satisfies $\mu \le \lambda$, that is, $\lambda -\mu$ is a nonnegative integral combination of the simple roots (Highest-weight classification, The highest-weight space is one-dimensional, Highest weight modules lie below the top weight, Root order on weights).

Weight multiplicities of $V(\lambda )$ are invariant under $W$: $\dim V(\lambda {)}_{w\mu }=\dim V(\lambda {)}_{\mu }$ for all $w\in W$ and all $\mu$, so the weight set is $W$-invariant (Simple reflections preserve weight multiplicities).

The chambers of $\mathrm{\Phi }$ are the connected components of the complement of the root hyperplanes; the fundamental chamber is $C=\{x:(x,{\alpha }_i)>0\}$, the Weyl group permutes the chambers, and $W$ acts simply transitively on them; the set of positive roots attached to $w(C)$ is $w({\mathrm{\Phi }}^{+})$, and the associated positive cone is $w(Q_{+})$ (Open and closed Weyl chambers, Simple transitivity on Weyl chambers, Simple roots form a signed integral basis, Weyl group).