Geometric q-character formula for real simple modules

Let $m$ be a dominant monomial in the variables $Y_{i,r}\in\mathbf{Y}^-$. Write

$$Y_{i,r}=\frac{z_{i,r}}{z_{i,r+b_{ii}}},\qquad ((i,r)\in W^-),$$

with $z_{i,s}=1$ for $s>0$, and define

$$m=\mathbf{z}^{g(m)}:=\prod_{(i,r)\in W^-}z_{i,r}^{g_{i,r}(m)}.$$

The integer vector $g(m)\in\mathbb{Z}^{(W^-)}$ is the $g$-vector of $L(m)$. Let $K(m)$ be the kernel of a generic homomorphism from $I(m)^-$ to $I(m)^+$, where

$$I(m)^+=\bigoplus_{g_{i,r}(m)>0}I_{i,r-d_i}^{\oplus g_{i,r}(m)},\qquad I(m)^-=\bigoplus_{g_{i,r}(m)<0}I_{i,r-d_i}^{\oplus |g_{i,r}(m)|}.$$

Let $F_{K(m)}$ be the associated $F$-polynomial, with its variables evaluated as specified in the source. Geometric q-character conjecture. If $L(m)$ is an irreducible real $U_q(\widehat{\mathfrak{g}})$-module in $\mathcal{C}^-$, then

$$\chi_q^-(L(m))=mF_{K(m)}.$$

This conjecture proposes a geometric formula for truncated $q$-characters of real simple modules, extending the geometric formulas established earlier in the paper; its general validity remains open in the source.