The graded Cartan pairing between projective and module classes

The pairing is defined and finite. Fix a root degree $\alpha$. Since $R_{\alpha }^{\mathrm{\Lambda }}$ is finite dimensional, a finite homogeneous generating family makes $P$ a quotient of a finite sum of shifted regular modules, hence finite dimensional. Each ${\mathrm{Hom}}_{R_{\alpha }^{\mathrm{\Lambda }},d}(P,M)$ is a subspace of the finite-dimensional space of $k$-linear maps from $P$ to $M$. The finite grading supports of $P$ and $M$ imply that this subspace vanishes unless $d\in \mathrm{supp}(M)-\mathrm{supp}(P)$. Thus the sum defining $h_{\alpha }(P,M)$ is a finite Laurent polynomial, including zero when either module is zero.

The pairing descends. Additivity of the graded Hom pairing in $P$ and $M$, and its behaviour under split exact sequences in $P$ and short exact sequences in $M$, are those of [F1]; therefore $h_{\alpha }$ descends to $K_0^{\mathrm{g}\mathrm{r}}(R_{\alpha }^{\mathrm{\Lambda }})\times G_0^{\mathrm{g}\mathrm{r}}(R_{\alpha }^{\mathrm{\Lambda }})$ and, by the restricted-support definition of the towers, to the displayed pairing on the restricted direct sums [F1].

The twist formula. Internal shift satisfies ${\mathrm{Hom}}_d(P\{r\},M)\cong {\mathrm{Hom}}_{d+r}(P,M)$, so $h_{\alpha }(P\{r\},M)=q^{-r}{h}_{\alpha }(P,M)$ and $h_{\alpha }(P,M\{s\})=q^s{h}_{\alpha }(P,M)$; hence $h(q^r[P],q^s[M])=q^{s-r}h([P],[M])$, the sesquilinear twist of [F1] [F1]. In particular $h^0(P,S)={\dim }_k{\mathrm{Hom}}^0(P,S)$ is the coefficient of $q^0$ in $h(P,S)$; evaluation at $q=1$ instead sums the dimensions in all degrees.

Transport and duality conventions. The transport isomorphisms of [F3] identify the Hecke-side groups with the KLR-side groups and carry internal shift to internal shift, so the pairing defined above transports to the Hecke side without change; the dualities $\psi$, $D_{\psi }$ and $\mathrm{\#}$ are shift-reversing involutions of [F3], and each graded-simple shift orbit has the unique $D_{\psi }$-self-dual representative of [F3], and its indecomposable projective cover has the normalized $\mathrm{\#}$-self-dual grading with ${\mathrm{Hom}}^0(P,S)=k$. These assertions concern simples and their indecomposable covers; they do not normalize every projective direct sum by an overall shift [F3]. No perfectness, nondegeneracy or $A$-self-duality is asserted at this point; those are the content of the integral highest weight identification named in the Definition.

No choice. Every ingredient is a finite-dimensional block, a finite Laurent polynomial or a transported isomorphism; the normalized representatives are selected canonically inside their orbits by the duality fixing condition, and no family of nonempty sets is selected. Hence no choice principle is used.