Restricted Chevalley dual

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 Chevalley-contravariant forms. 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.