The projective-character submodule conjecture for the torus representation

Let $\bar U_q$ be the restricted quantum group, let $\operatorname{SLF}(\bar U_q)$ be its space of symmetric linear forms, and let $\mathcal{L}_{1,0}^{\mathrm{inv}}(\bar U_q)$ act on this space. Define

$$\mathcal{P}=\operatorname{vect}\bigl(\chi^+_s+\chi^-_{p-s},\chi^\pm_p\bigr)_{1\leq s\leq p-1}.$$

Projective-character submodule conjecture. The subspace $\mathcal{P}$ is an $\mathcal{L}_{1,0}^{\mathrm{inv}}(\bar U_q)$-submodule of $\operatorname{SLF}(\bar U_q)$. The subspace is spanned by characters of projective $\bar U_q$-modules, and the conjecture asks whether it remains stable under the full invariant-loop algebra rather than only the smaller subalgebra whose action is analyzed later in the paper.