The KNO conjecture on geometric crystals for fundamental representations

Let $\mathfrak g$ be an affine Lie algebra, $G$ its affine Kac–Moody group, and $I$ its index set with distinguished affine node $0$. Let $W(\varpi_i)$ be the Kirillov–Reshetikhin module with extremal weight $\varpi_i$, let $\mathbb P(\varpi_i)$ be its projective space, and let $c_i^\vee$ be as above. For $J\subset I$, write $\mathfrak g_J$ for the subalgebra generated by $\{e_j,f_j\}_{j\in J}$, and for an integral weight $\mu$ set

$$I(\mu):=\{j\in I\mid\langle\alpha_j^\vee,\mu\rangle\geq0\}.$$

For a reduced word $w=s_{i_1}\cdots s_{i_k}$, let $B^-_{i_1,\ldots,i_k}$ be the corresponding Schubert-cell chart, and let $\overline u$ denote the line in $\mathbb P(\varpi_i)$ containing an extremal vector $u$.

KNO conjecture. For any $i\in I\setminus\{0\}$, there exists a unique variety $X$ endowed with a positive $\mathfrak g$-geometric crystal structure and a rational map $\pi:X\to\mathbb P(\varpi_i)$ such that, for every extremal vector $u\in W(\varpi_i)_\mu$, if $t(c_i^\vee\mu)=\tau w$ with $\tau$ a Dynkin diagram automorphism and $w=s_{i_1}\cdots s_{i_k}$, then there is a birational map $\xi:B^-_{i_1,\ldots,i_k}\to X$ that is a morphism of $\mathfrak g_{I(\mu)}$-geometric crystals and

$$\pi\circ\xi\left(Y_{i_1}(x_1)\cdots Y_{i_k}(x_k)\right)=Y_{i_1}(x_1)\cdots Y_{i_k}(x_k)\overline u,$$

while the ultra-discretization of $X$ is isomorphic to the crystal $B_\infty=B_\infty(\varpi_i)$ of the Langlands dual $\mathfrak g^L$.

The conjecture seeks a unique geometric-crystal model whose projective realization is compatible with extremal vectors and whose ultra-discretization yields the limiting crystal for the Langlands dual algebra. The source gives no evidence of a resolution.