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For the sl2 Lie algebra, let $v_{\lambda }$ be the highest-weight vector. The Poincare-Birkhoff-Witt theorem gives the basis $(f^r{v}_{\lambda }{)}_{r\ge 0}$, and the defining Lie brackets implyA positive-degree basis vector is singular exactly when $r=\lambda +1$ is a positive integer. Therefore $M_{\lambda }$ is irreducible when $\lambda \notin {\mathbb{Z}}_{\ge 0}$.

If $\lambda =m\in {\mathbb{Z}}_{\ge 0}$, the vector $f^{m+1}{v}_{\lambda }$ has weight $-m-2$ and generates a submodule isomorphic to $M_{-m-2}$. The latter is irreducible because $-m-2\notin {\mathbb{Z}}_{\ge 0}$. Every nonzero submodule contains a singular vector by the preceding part, and the displayed coefficient shows that this is the only possible proper singular vector. Henceis the unique proper nonzero submodule, as summarized by the Reducibility of an sl2 Verma module.

Solved by gpt-5.6-sol high.

The 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 $M_{\lambda }$ byThe same element acts on the highest-weight vector $v$ of weight $\mu$ byBoth are the action of one operator on the same module, so the scalars agree and

Solved by gpt-5.6-sol high.