Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 102 / 4 / d / Solution

The Poincare-Birkhoff-Witt theorem shows that the weights of the Verma module $M({\omega }_2)$ arewith multiplicities given by the corresponding Kostant partition function.

The Dynkin labels of ${\omega }_2$ are $(0,1)$. Hence the two simple-root singular vectors are $f_1{v}_{{\omega }_2}$, of weight ${\omega }_2-{\alpha }_1$, and $f_2^2{v}_{{\omega }_2}$, of weight ${\omega }_2-2{\alpha }_2$. They generate the Maximal proper submodule of a dominant integral Verma module. Its set of weights is consequently