= Solution
Let $\mathfrak g$ be a finite-dimensional complex <semisimple Lie algebra>, choose a <Cartan subalgebra> $\mathfrak t$ and a set $R^+$ of <positive roots>, and write $W$ for the <Weyl group>. Let $\lambda$ be a <dominant integral weight> and $L(\lambda)$ the finite-dimensional irreducible <highest-weight representation> with that highest weight. If $L(\lambda)_\mu$ denotes its <weight space> of weight $\mu$, its <formal character> is $\operatorname{ch}L(\lambda)=\sum_\mu\dim L(\lambda)_\mu\,e^\mu$. The symbols $e^\mu$ belong to the <group algebra> of the <weight lattice> and multiply by $e^\mu e^\nu=e^{\mu+\nu}$; they are formal symbols, not matrix exponentials.
Put $\rho=\frac12\sum_{\alpha\in R^+}\alpha$, the <half-sum of positive roots>, and let $\ell(w)$ be the <Coxeter length> of $w$, the least number of simple reflections expressing it. Then $(-1)^{\ell(w)}=\det(w)$ in the real reflection representation. The <Weyl character formula> states
$$
\boxed{\operatorname{ch}L(\lambda)=\frac{\displaystyle\sum_{w\in W}(-1)^{\ell(w)}e^{w(\lambda+\rho)}}{\displaystyle\sum_{w\in W}(-1)^{\ell(w)}e^{w\rho}}.}
$$
By the <Weyl denominator formula>, the denominator is $e^\rho\prod_{\alpha\in R^+}(1-e^{-\alpha})$, giving the equivalent expression
$$
\boxed{\operatorname{ch}L(\lambda)=\frac{\displaystyle\sum_{w\in W}\det(w)e^{w(\lambda+\rho)-\rho}}{\displaystyle\prod_{\alpha\in R^+}(1-e^{-\alpha})}.}
$$
The quotient can be interpreted in the fraction field of the formal group algebra; the theorem says it simplifies to the finite <formal character> above. In particular the apparent denominator does not mean that the representation has infinitely many weights.
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