Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 2 / 3 / ii / Solution

For $v\in V_{\lambda }$ and $w\in W_{\mu }$, the tensor product of Lie algebra representations acts byThus its weight space of weight $\nu$ isSubstituting this into the formal character and using $e(\lambda +\mu )=e(\lambda )e(\mu )$ provesEvery sum is finite for the Lie algebra representations under consideration.