Restricted self-duality of simple highest-weight modules

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 $w\cdot \lambda =w(\lambda +\rho )-\rho$. For an $\mathfrak{h}$-semisimple module $M$ with finite-dimensional weight spaces, its restricted Chevalley dual is $D(M)={⨁}_{\mu \in {\mathfrak{h}}^{\ast }}{M}_{\mu }^{\ast },(x\phi )(m)=\phi (\tau (x)m)(x\in U(\mathfrak{g}))\mathrm{.}$ Here each functional is extended by zero on the other weight spaces and $\tau$ is the fixed anti-involution of def-chevalley-contravariant-form. In particular $\tau (h)=h$ and $D(M{)}_{\mu }=M_{\mu }^{\ast }$. A map $f:M\to N$ induces $D(f):D(N)\to D(M)$ by precomposition. The action law follows from $\tau (xy)=\tau (y)\tau (x)$; a root vector of weight $\alpha$ sends $M_{\mu }^{\ast }$ to $M_{\mu +\alpha }^{\ast }$, so the restricted sum is stable. This is a complex-linear algebraic dual, with no conjugation. Ordinary Lie-module duality has a minus sign and reverses weights; twisting that dual by the Lie automorphism $x\mapsto -\tau (x)$ gives the convention used here. (Restricted Chevalley dual)

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)

The weights of $M(\lambda )$ are exactly $\lambda -\beta$ for $\beta \in Q^{+}$; every weight space is finite dimensional, and $M(\lambda {)}_{\lambda }=\mathbb{C}{v}_{\lambda }$. (Weights of a Verma module lie below lambda)

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)