= Solution
With the source's negative <root> partition function, interpret $Q_+=\sum_{\alpha\in B}\mathbb Z_{\ge0}\alpha$ and allow zero coefficients in the partitions. Write $e^\lambda$ for $\delta_\lambda$, with multiplication meaning <convolution>. Then
$$
K=\prod_{\gamma>0}\left(\sum_{n\ge0}e^{-n\gamma}\right).
$$
The denominator needed for the asserted character identities is
$$
D=\prod_{\gamma>0}(e^{\gamma/2}-e^{-\gamma/2})
=e^\rho\prod_{\gamma>0}(1-e^{-\gamma}).
$$
The repeated identical subscripts in the printed denominator would instead make it zero, so those literal factors cannot satisfy an inverse identity. Also the required inverse uses the negative $\rho$ shift. Geometric-series cancellation gives
$$
\boxed{K*\delta_{-\rho}*D=\delta_0.}
$$
With a positive shift the left side would be $\delta_{2\rho}$, which displays the second printed sign problem even after the denominator is repaired.
<Poincaré-Birkhoff-Witt theorem> gives $\pi_{M^\xi}=\delta_{\xi-\rho}*K$, since each ordered negative <root> monomial lowers the <highest weight> by its corresponding partition. Therefore
$$
\boxed{D*\pi_{M^\xi}=\delta_\xi.}
$$
The remaining parts use $D$ for the intended <Weyl denominator>, while the private TeX preserves the printed signs.
Back to article page
