Irreducible quotient of a Verma module

The sum of all proper submodules of $M_{\lambda }$ is its unique maximal proper submodule. The quotient is the unique irreducible highest-weight module $V(\lambda )$.

If $\lambda$ is a dominant integral weight, the maximal proper submodule of $M(\lambda )$ is generated by the simple-root singular vectorsThe generator associated with ${\alpha }_i$ has weight $s_i(\lambda +\rho )-\rho$.