Kostant's weight multiplicity formula

Given: The Axiom of Choice, a dominant integral weight $\lambda \in {\mathrm{\Lambda }}^{+}$, an element $\mu \in {\mathfrak{h}}^{\ast }$, the multiplicities $m_{\lambda }(\mu )$ of $L(\lambda )$, the Kostant partition function $P$ and the completed character ring $\mathcal{R}$.

The Axiom of Choice is assumed; it enters through the Weyl character formula of [F1] (The Axiom of Choice).

$\mathrm{ch}L(\lambda )=A(\lambda +\rho )\cdot A(\rho {)}^{-1}$ with $A(\lambda +\rho )={\sum }_{w\in W}(-1{)}^{\mathrm{\ell }(w)}{e}^{w(\lambda +\rho )}$ and $A(\rho {)}^{-1}=e^{-\rho }{\prod }_{\alpha \in {\mathrm{\Phi }}^{+}}(1-e^{-\alpha }{)}^{-1}$ (The Weyl character formula, The Weyl denominator identity, The Weyl alternation operator, Geometric series are invertible in the completed character ring).

${\prod }_{\alpha \in {\mathrm{\Phi }}^{+}}(1-e^{-\alpha }{)}^{-1}={\sum }_{\beta \in Q_{+}}P(\beta )e^{-\beta }$ in $\mathcal{R}$, with $P(\beta )\in {\mathbb{Z}}_{\ge 0}$ finite and $P(\nu )=0$ for $\nu \mathrm{\notin }{Q}_{+}$ (The Kostant partition function).

$\mathrm{ch}L(\lambda )={\sum }_{\mu }{m}_{\lambda }(\mu )e^{\mu }$, so $m_{\lambda }(\mu )$ is the coefficient of $e^{\mu }$ in $\mathrm{ch}L(\lambda )$ (The formal character of a finite-dimensional weight module).

In $\mathcal{R}$ the coefficient of $e^{\eta }$ in a product is the finite sum ${\sum }_{\gamma +\delta =\eta }{c}_{\gamma }{d}_{\delta }$ of the coefficients of the factors, and coefficient extraction is additive over finite sums (The completed formal character ring).