= Solution
For $v\in V_\lambda$ and $w\in W_\mu$, the <tensor product of Lie algebra representations> acts by
$$
H(v\otimes w)=Hv\otimes w+v\otimes Hw=(\lambda(H)+\mu(H))v\otimes w.
$$
Thus its <weight space> of <weight> $\nu$ is
$$
(V\otimes W)_\nu=\bigoplus_{\lambda+\mu=\nu}V_\lambda\otimes W_\mu,
\qquad
\dim(V\otimes W)_\nu=\sum_{\lambda+\mu=\nu}(\dim V_\lambda)(\dim W_\mu).
$$
Substituting this into the <formal character> and using $e(\lambda+\mu)=e(\lambda)e(\mu)$ proves
$$
\boxed{\operatorname{char}(V\otimes W)
=(\operatorname{char}V)(\operatorname{char}W).}
$$
Every sum is finite for the <Lie algebra representations> under consideration.
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