= Solution
Choose the <Borel subalgebra> $\mathfrak b=\mathfrak t\oplus\mathfrak n^+$ determined by the positive roots. Regard the one-dimensional space $\mathbb C_\lambda$ as a $\mathfrak b$-module on which $\mathfrak n^+$ acts by zero and $h\in\mathfrak t$ acts by $\lambda(h)$. The <Verma module> is
$$
M_\lambda=U(\mathfrak g)\otimes_{U(\mathfrak b)}\mathbb C_\lambda.
$$
The <Poincare-Birkhoff-Witt theorem> identifies it as a vector space with $U(\mathfrak n^-)$ acting on a highest-weight vector $v_\lambda$.
For each positive root $\alpha$, arbitrary powers of a negative-root vector contribute the geometric series $1+e^{-\alpha}+e^{-2\alpha}+\cdots$. Consequently the <formal character of a weight module> is
$$
\operatorname{ch}M_\lambda
=e^\lambda\prod_{\alpha\in R^+}(1-e^{-\alpha})^{-1}.
$$
This product is interpreted in the completion of the group algebra in the negative-root direction; the PBW basis proves that every coefficient is the correct finite <weight space> dimension.
Solved by gpt-5.6-sol high.
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