Simple extensions cannot cross linkage classes

Given: The Axiom of Choice (The Axiom of Choice), the setting above and the hypotheses in the statement.

Under the Axiom of Choice, fix a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$, a Cartan subalgebra $\mathfrak{h}$, and a positive Borel $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$. Write $Q^{+}={\sum }_i{\mathbb{Z}}_{\ge 0}{\alpha }_i$, $\mu \le \lambda$ when $\lambda -\mu \in Q^{+}$, and $w\cdot \lambda =w(\lambda +\rho )-\rho$. Restricted Chevalley duality is an exact contravariant equivalence $D:\mathcal{O}\to {\mathcal{O}}^{\mathrm{o}\mathrm{p}}$, with a natural isomorphism $D^2\cong \mathrm{id}$. It preserves each weight-space dimension, the formal character, and every simple composition multiplicity. (Restricted duality is exact and involutive on O)

For a $\mathfrak{g}$-module $V$, sending a homomorphism $T:M(\lambda )\to V$ to $T(v_{\lambda })$ is a bijection onto the vectors $v\in V$ of weight $\lambda$ annihilated by ${\mathfrak{n}}^{+}$. Here $M(\lambda )$ is def-verma-module. The nonzero vectors in this target are precisely the highest-weight vectors of weight $\lambda$ from def-highest-weight-vector-and-cyclic-highest-weight-module; the zero vector corresponds to the zero homomorphism. (The universal property of Verma modules)

Under the Axiom of Choice, if $[M(\lambda ):L(\mu )]\mathrm{\ne }0$, then $\mu \uparrow \lambda$. (The strong linkage principle for Verma modules)

Fix a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$, a Cartan subalgebra $\mathfrak{h}$, and a positive Borel $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$. Write $Q^{+}={\sum }_i{\mathbb{Z}}_{\ge 0}{\alpha }_i$, $\mu \le \lambda$ when $\lambda -\mu \in Q^{+}$, and $w\cdot \lambda =w(\lambda +\rho )-\rho$. The set ${\mathrm{\Phi }}_{\lambda }$ is preserved by its root reflections. If $\mu \in W_{\lambda }\cdot \lambda$, then ${\mathrm{\Phi }}_{\mu }={\mathrm{\Phi }}_{\lambda }$ and $W_{\mu }=W_{\lambda }$. The equivalence classes generated by moves $\eta \mapsto s_{\alpha }\cdot \eta$ with $\alpha \in {\mathrm{\Phi }}_{\eta }$ are exactly $W_{\lambda }\cdot \lambda$. Every strong-linkage chain stays in one such class. (Integral reflection linkage is an equivalence relation)

The proper submodule $J(\lambda )$ which is the sum of all proper submodules is the unique maximal submodule of $M(\lambda )$. The quotient $L(\lambda ):=M(\lambda )\mathrm{/}J(\lambda )$ is simple and is its unique simple quotient. (A Verma module has a unique simple quotient)