The cyclotomic module $G_0$ is the restricted dual integral highest-weight module

Perfectness and the dual isomorphism. By [F1] each weight component has finitely many absolutely simple graded shift orbits and normalized projective covers, and the exactness of projective tensor pairing together with the unscaled diagonal values gives perfectness over $A$ on each weight. Hence $G_0$ is free on the simple classes and the functional map ${\mathrm{\Psi }}_{\mathrm{c}\mathrm{a}\mathrm{n}}(M)(\varphi (P))=T(P,M)$ is an isomorphism from $G_0(R^{\mathrm{\Lambda }})$ to the weightwise dual ${⨁}_{\beta }{\mathrm{Hom}}_A(V_A(\mathrm{\Lambda }{)}_{\mathrm{\Lambda }-\beta },A)$, i.e. the restricted weight dual and not a full unrestricted product dual.

Intertwining with the canonical action. Transport $K_0$ by $\varphi$; its tensor form is the actual highest-weight contravariant form with $\sigma (F_i)=q_i^{-1}{K}_i{E}_i$ (The contravariant form on an integrable highest weight module). From tensor adjunction, $T(P,F_i^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}M)=T(E_i^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}P,M)=T(\sigma (F_i)P,M)$, and the analogous normalized identity is $T(P,E_i^{\mathrm{c}\mathrm{a}\mathrm{n}}M)=T(\sigma (E_i)P,M)$; these are precisely the restricted contravariant-dual action relations, so ${\mathrm{\Psi }}_{\mathrm{c}\mathrm{a}\mathrm{n}}$ intertwines the canonical module action with the dual action and sends normalized simple classes to the form-dual projective basis.

The KK normalization and the Hom pairing. The Kang--Kashiwara action differs from the canonical one by the weight rescaling $q^{{\mathrm{def}}_{\mathrm{\Lambda }}(\beta )}$ on root $\beta$, so its intertwiner is ${\mathrm{\Psi }}_{\mathrm{K}\mathrm{K}}{\mathrm{\mid }}_{\beta }=q^{{\mathrm{def}}_{\mathrm{\Lambda }}(\beta )}{\mathrm{\Psi }}_{\mathrm{c}\mathrm{a}\mathrm{n}}{\mathrm{\mid }}_{\beta }$; in particular the unrescaled KK identification is not asserted to send simples to the upper global basis. The Hom pairing satisfies $h(P,M)=T(P^{\mathrm{\#}},M)$ by [F1] and is first-bar-sesquilinear, which concerns the other duality and does not alter the bilinear tensor-dual identification. The Axiom of Choice is inherited only through the integral projective identification [F2] and the contravariant-form suppliers [F3].