= Solution
The <weight of a representation>[weights] of $M_\lambda$ lie below $\lambda$ in the positive-root order, and every <weight space> is finite-dimensional. A nonzero submodule $V$ is stable under the <Cartan subalgebra>, so it is a direct sum of its weight spaces. Choose a maximal weight $\mu$ occurring in $V$. Every positive-root operator would raise its weight; maximality therefore makes it kill any nonzero $v\in V_\mu$. Thus $v$ is a <singular vector>.
The <Casimir element> is central and acts throughout $M_\lambda$ by
$$
|\lambda+\rho|^2-|\rho|^2.
$$
The same element acts on the highest-weight vector $v$ of weight $\mu$ by
$$
|\mu+\rho|^2-|\rho|^2.
$$
Both are the action of one operator on the same module, so the scalars agree and
$$
|\mu+\rho|=|\lambda+\rho|.
$$
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