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

With the source's negative root partition function, interpret $Q_{+}={\sum }_{\alpha \in B}{\mathbb{Z}}_{\ge 0}\alpha$ and allow zero coefficients in the partitions. Write $e^{\lambda }$ for ${\delta }_{\lambda }$, with multiplication meaning convolution. ThenThe denominator needed for the asserted character identities isThe 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 givesWith 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 ${\pi }_{M^{\xi }}={\delta }_{\xi -\rho }\ast K$, since each ordered negative root monomial lowers the highest weight by its corresponding partition. ThereforeThe remaining parts use $D$ for the intended Weyl denominator, while the private TeX preserves the printed signs.