Past exam of the representation theory of semisimple Lie algebras course of the University of Oxford / 2010 / 3 / d / Solution

Convolve the identity of part (c) with the inverse denominator $K\ast {\delta }_{-\rho }$ found in part (a). Associativity givesTranslation by a Kronecker delta obeys $(K\ast {\delta }_{\lambda })(\psi )=K(\psi -\lambda )$. Taking the coefficient at $\psi$ therefore yieldsThis is the Kostant multiplicity formula in the paper's convention that $K$ counts negative root partitions. With a partition function counting positive root sums, each argument would have the opposite sign.