= Solution
Convolve the identity of part (c) with the inverse denominator $K*\delta_{-\rho}$ found in part (a). Associativity gives
$$
\pi_{A^\phi}=\sum_{w\in W}(\det w)\,K*\delta_{w(\phi+\rho)-\rho}.
$$
Translation by a <Kronecker delta> obeys $(K*\delta_\lambda)(\psi)=K(\psi-\lambda)$. Taking the coefficient at $\psi$ therefore yields
$$
\boxed{\pi_{A^\phi}(\psi)=\sum_{w\in W}(\det w)
K\bigl(\psi+\rho-w(\phi+\rho)\bigr).}
$$
This 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.
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