The formal character of a finite-dimensional weight module

Assume the Axiom of Choice (The Axiom of Choice). Let $V$ be a finite-dimensional $\mathfrak{g}$-module, where $\mathfrak{g}$ is a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$, and let $V={⨁}_{\mu \in {\mathfrak{h}}^{\ast }}{V}_{\mu }$ be its weight-space decomposition, with $V_{\mu }$ the weight space of Weight and weight space (Finite-dimensional modules decompose into weight spaces). The formal character of $V$ is $\mathrm{ch}V:={\sum }_{\mu \in {\mathfrak{h}}^{\ast }}(\dim V_{\mu })e^{\mu }\in \mathcal{R},$ the finite sum with integer coefficients taken in the completed formal character ring $\mathcal{R}$ of The completed formal character ring; equivalently, $\mathrm{ch}V$ is the coefficient family $\mu \mapsto \dim V_{\mu }$ on ${\mathfrak{h}}^{\ast }$, which has finite support by the cited decomposition.

We write $m_{\lambda }(\mu ):=\dim L(\lambda {)}_{\mu }$ for the multiplicity of $\mu$ as a weight of the finite-dimensional simple module $L(\lambda )$ of highest weight $\lambda \in {\mathrm{\Lambda }}^{+}$ (Highest-weight classification), so that $\mathrm{ch}L(\lambda )={\sum }_{\mu }{m}_{\lambda }(\mu )e^{\mu }$; the coefficients $m_{\lambda }(\mu )$ are nonnegative integers and $m_{\lambda }(\mu )=0$ for all but finitely many $\mu$.