Weyl alternation extracts a dominant highest-weight coefficient

Given: AC, a finite-dimensional $\mathfrak{g}$-module $V$ with decomposition $V\cong {⨁}_{\mu \in {\mathrm{\Lambda }}^{+}}L(\mu {)}^{\oplus [V:L(\mu )]}$ and a dominant integral weight $\nu$.

Weyl character formula: $\mathrm{ch}L(\mu )=A(\mu +\rho )\mathrm{/}A(\rho )$ for $\mu \in {\mathrm{\Lambda }}^{+}$, i.e. $A(\rho )\mathrm{ch}L(\mu )=A(\mu +\rho )$; the alternants have finite support, the completed ring $\mathcal{R}$ contains $A(\rho )$ as an invertible element with inverse the Weyl-denominator geometric series, and $A(\eta )={\sum }_{w\in W}(-1{)}^{\mathrm{\ell }(w)}{e}^{w\eta }$ (The Weyl character formula, The Weyl alternation operator, Geometric series are invertible in the completed character ring, The completed formal character ring).

The formal character is additive over direct sums and multiplicative over tensor products, and it determines the multiplicities: the coefficient of $\mathrm{ch}L(\mu )$ in $\mathrm{ch}V={\sum }_{\mu }{a}_{\mu }\mathrm{ch}L(\mu )$ satisfies $a_{\mu }=[V:L(\mu )]$ (Formal characters are additive and multiplicative, Tensor-product multiplicities are character structure constants, Weyl's complete reducibility theorem, Highest-weight classification).

For each $\nu \in {\mathrm{\Lambda }}^{+}$, $\nu +\rho$ is strictly dominant (Positive coroot pairings of a dominant integral weight). Each real Weyl orbit has one closed-dominant representative, and the stabilizer of that representative is generated by the simple reflections whose walls contain it. Thus the stabilizer of $\nu +\rho$ is trivial, and if $w(\mu +\rho )=\nu +\rho$ for dominant integral $\mu ,\nu$, uniqueness first gives $\mu =\nu$, then triviality of the stabilizer gives $w=1$ (Finite Weyl closed chambers and stabilizers, Integral, dominant, and strictly dominant weights, Finite Weyl root system, lattice and chamber conventions).