Kostant multiplicity formula ($m_{\lambda }(\mu )={\sum }_{w\in W}\epsilon (w)p(w(\lambda +\rho )-(\mu +\rho ))$)

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.