The simple socle of a costandard object

Given: The setting above and the hypotheses in the statement.

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)

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)

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 $w\cdot \lambda =w(\lambda +\rho )-\rho$. For $\lambda \in {\mathfrak{h}}^{\ast }$, the standard and costandard objects are $\mathrm{\Delta }(\lambda )=M(\lambda ),\mathrm{\nabla }(\lambda )=D(M(\lambda ))\mathrm{.}$ The Verma module is defined in def-verma-module, and prop-restricted-duality-is-an-exact-involution-on-category-o supplies the duality on $\mathcal{O}$. These symbols name the two objects; no projectivity or highest-weight-category axiom is part of this definition. (Standard and costandard objects)