The <Poincare-Birkhoff-Witt theorem> shows that the weights of the <Verma module> $M(\omega_2)$ are
$$
\boxed{\omega_2-Q_+
=\{\omega_2-n_1\alpha_1-n_2\alpha_2:n_1,n_2\in\mathbb Z_{\geq0}\},}
$$
with 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 $f_1v_{\omega_2}$, of weight $\omega_2-\alpha_1$, and $f_2^2v_{\omega_2}$, 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
$$
\boxed{(\omega_2-\alpha_1-Q_+)\ \cup\ (\omega_2-2\alpha_2-Q_+).}
$$

Solution

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

{c}

If $\lambda$ is a <dominant integral weight>, the maximal proper submodule of $M(\lambda)$ is generated by the simple-root <singular vectors>
$$
f_i^{\langle\lambda,\alpha_i^\vee\rangle+1}v_\lambda.
$$
The generator associated with $\alpha_i$ has weight $s_i(\lambda+\rho)-\rho$.

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 Incoming links: Maximal proper submodule of a dominant integral Verma module