The upper (form-dual) global basis and the required perfectness of the integral pairing

Let $M={⨁}_{\mu }{M}_{\mu }$ be $U_q^{-}$ or an integrable highest weight module $V(\lambda )$, with finite root or label components, lower global $A$-lattice $M_A={⨁}_{\mu }(M_A{)}_{\mu }={⨁}_{b\in \mathcal{B}}AG(b)$ of The lower global basis of $U_q^{-}$ and of a highest weight module with rational lower charts $L_0$, $L_{\mathrm{\infty }}=\psi (L_0)$ of the balanced triple Balanced triples over the two charts of ${\mathbb{P}}^1$ and the globalization map, and let $B$ be the normalized homogeneous Kashiwara half form, respectively the contravariant polarization of The Lusztig-Kashiwara bilinear form on the negative half: the recursion, The negative-half bilinear form is symmetric, nondegenerate and integral and The contravariant form on an integrable highest weight module.

The restricted graded integral form-dual of $M_A$ is $M_A^{\vee }:={⨁}_{\mu }\{x\in M_{\mu }:B(x,(M_A{)}_{\mu })\subseteq A\},$ the direct sum being taken over weight components so that ordinary (not product) duals of the infinite-dimensional pieces are used; the dual rational charts are $L_0^{\vee }:={⨁}_{\mu }\{x\in M_{\mu }:B(x,(L_0{)}_{\mu })\subseteq A_0\},L_{\mathrm{\infty }}^{\vee }:={⨁}_{\mu }\{x\in M_{\mu }:B(x,(L_{\mathrm{\infty }}{)}_{\mu })\subseteq A_{\mathrm{\infty }}\}\mathrm{.}$ The upper global family is the unique homogeneous field family $G^{\mathrm{u}\mathrm{p}}(b)$ ($b\in \mathcal{B}$) determined by the inverse Gram system $B(G(b^{\mathrm{\prime }}),G^{\mathrm{u}\mathrm{p}}(b))={\delta }_{b,b^{\mathrm{\prime }}}(b,b^{\mathrm{\prime }}\in {\mathcal{B}}_{\mu });$ it is defined componentwise by finite-dimensional linear algebra because each component form is nondegenerate. The upper bar ${\psi }^{\vee }$ is the $B$-adjoint semilinear involution characterized by $B(\psi u,{\psi }^{\vee }v)=\overline{B(u,v)}$, and the upper Kashiwara operators are the adjoints ${\tilde{e}}_i^{\mathrm{u}\mathrm{p}}:=({\tilde{f}}_i^{\mathrm{l}\mathrm{o}\mathrm{w}}{)}^{\mathrm{a}\mathrm{d}\mathrm{j}}$ and ${\tilde{f}}_i^{\mathrm{u}\mathrm{p}}:=({\tilde{e}}_i^{\mathrm{l}\mathrm{o}\mathrm{w}}{)}^{\mathrm{a}\mathrm{d}\mathrm{j}}$, defined componentwise by the same nondegenerate finite-dimensional forms; they are in general distinct from the lower operators.

The upper zero chart is the dual chart $L_0^{\vee }$, and after the grand loop its perfection identifies it with $L_0$; the upper infinity chart $L_{\mathrm{\infty }}^{\vee }$ is not automatically the lower bar chart $L_{\mathrm{\infty }}$, and no such identification is asserted. Existence of the upper family in the integral lattices, weightwise perfect $A$-duality, dual globalization in these genuine dual charts and crystal compatibility for the adjoint operators are the exact conclusions of Upper and lower global bases are dual and crystal compatible; adjacent divided-string formulas of the next lemma supply the adjoints of the lower string shifts with their weights, so no unweighted adjunction, no full product identification $M_A^{\vee }={\mathrm{Hom}}_A(M_A,A)$, no lower $A$-self-duality, no ordinary-bar fixedness and no unqualified upper/lower operator equality is assumed here.