The full toral-weight Lusztig module category

Fix a simple simply-laced finite root system, its full weight lattice $P$ and coroot lattice $Q^{\vee }$, with the simply connected root datum $X=P$, $Y=Q^{\vee }$ and $d_i=1$. Let $\zeta \in {\mathbb{C}}^{\times }$ satisfy $\mathrm{ord}({\zeta }^2)=\mathrm{\ell }$, where $\mathrm{\ell }>h$ is odd and $h$ is the Coxeter number. Both orders $\mathrm{\ell }$ and $2\mathrm{\ell }$ for $\zeta$ are allowed. Use the full Lusztig form of De Concini–Kac, Lusztig, and small quantum-group forms at roots of unity with this datum, specialized at $q=\zeta$.

Let ${\mathcal{C}}_{\zeta }$ consist of finite-dimensional left modules with a finite $P$-weight decomposition $M={⨁}_{\mu \in P}{M}_{\mu }$ satisfying, for every $h\in Q^{\vee }$, $i\in I$, $r,t\ge 0$ and $c\in \mathbb{Z}$,

$K_h{\mathrm{\mid }}_{M_{\mu }}={\zeta }^{⟨\mu ,h⟩}\mathrm{id},T_{i,c,t}{\mathrm{\mid }}_{M_{\mu }}={{\left[\begin{array}{c}⟨\mu ,h_i⟩+c\\ t\end{array}\right]}_q\mid }_{q=\zeta }\mathrm{id},$

$E_i^{(r)}{M}_{\mu }\subseteq M_{\mu +r{\alpha }_i},F_i^{(r)}{M}_{\mu }\subseteq M_{\mu -r{\alpha }_i}\mathrm{.}$

The binomial is evaluated as a Laurent polynomial, including negative upper indices, rather than as a ratio of specialized factorials. Morphisms preserve weights and commute with the full algebra. The structure includes all divided powers and toral binomials; ordinary-generator eigenvalue conditions alone are insufficient at a root.

Normalize the symmetric rational bilinear form on $P$ by $({\alpha }_i,{\alpha }_j)=a_{ij}$. Choose a positive integer $D$ with $D(P,P)\subseteq \mathbb{Z}$ and choose $t\in {\mathbb{C}}^{\times }$ with $t^D=\zeta$. Define the symmetric bicharacter

$\chi (\lambda ,\mu )=t^{D(\lambda ,\mu )}\mathrm{.}$

This is part of the choice of braiding. In particular $\chi ({\alpha }_i,\mu )={\zeta }^{⟨\mu ,h_i⟩}$ on the entire selected weight lattice. No coprimality of $\mathrm{\ell }$ and the determinant of the Cartan matrix is imposed.