Duality preserves linkage blocks and block orthogonality

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 $w\cdot \lambda =w(\lambda +\rho )-\rho$. For a linkage class $C=W_{\lambda }\cdot \lambda$, let ${\mathcal{O}}_C$ be the full subcategory of objects all of whose simple composition factors have labels in $C$. Then $\mathcal{O}={⨁}_C{\mathcal{O}}_C$, and each nonzero ${\mathcal{O}}_C$ is indecomposable as a categorical direct summand. These are precisely the blocks. Each ${\mathcal{O}}_C$ lies in ${\mathcal{O}}_{{\chi }_{\lambda }}$; a central-character summand can contain several blocks. Independently, grouping weights by cosets of the root lattice $Q$ gives a canonical coarser decomposition by weight cosets. (Central-character summands refine into linkage blocks)

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)

Fix simple roots $\{{\alpha }_i\}$ in the chosen positive system and normalized Chevalley generators $e_i\in {\mathfrak{g}}_{{\alpha }_i}$, $f_i\in {\mathfrak{g}}_{-{\alpha }_i}$. Let $\tau :U(\mathfrak{g})\to U(\mathfrak{g})$ be the Chevalley anti-involution determined by $\tau (e_i)=f_i$, $\tau (f_i)=e_i$, and $\tau (h)=h$ for $h\in \mathfrak{h}$. A bilinear form $B$ on a $\mathfrak{g}$-module is Chevalley-contravariant when $B(xu,v)=B(u,\tau (x)v)(x\in U(\mathfrak{g}))\mathrm{.}$ This is a bilinear condition, not a Hermitian or positivity condition. (Chevalley-contravariant forms)