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Multiply the Weyl character formula by the generating series of the Kostant partition function. In each numerator summand, the coefficient of $e^{\mu }$ occurs when $\nu =w(\lambda +\rho )-(\mu +\rho )$. Summing those coefficients with the Weyl group signs proves the displayed weight multiplicity formula. Although $\rho$ need not itself be an integral group weight, $w\rho -\rho$ is in the root lattice, so each partition-function argument is an integral weight.

Use the corrected generating series of the Kostant partition function and the permitted Weyl character formula:For a fixed $w$, the coefficient of $e^{\mu }$ arises exactly whenThere are only finitely many Weyl-group terms, and the partition function vanishes outside the cone of nonnegative root combinations. Coefficient extraction is therefore legitimate in the formal completion used in the preceding part. Summing the signed contributions provesThis proves the Kostant multiplicity formula for every compact connected Lie group, not just the unitary groups. In the presence of a central torus, roots have zero central component, so the partition function automatically gives zero for any incompatible central weight. Also $\rho$ need not itself be an integral weight for the given global group: $w\rho -\rho$ always belongs to the root lattice, so every argument in the formula is nevertheless integral whenever $\lambda$ and $\mu$ are integral group weights.

As a normalization check, for $SU(2)$ write a highest weight as the nonnegative integer $\ell$, with positive root $2$ and $\rho =1$. The two Weyl-group terms giveSince $p(a)=1$ precisely when $a$ is even and nonnegative, this is one for $\mu =\ell ,\ell -2,\dots ,-\ell$ and zero otherwise, including the cancellation below the lowest weight. This recovers the full familiar weight string.