Canonical left tilde-S-region conjecture on completely prime ideals

Let ${^L \mathfrak{g}}$ be the Langlands-dual Lie algebra with Cartan subalgebra ${^L \mathfrak{h}}$, and identify ${^L \mathfrak{h}}^*$ with $\mathfrak{h}$. A canonical left $\tilde{S}$-region is the closure associated with a canonical left $\tilde{S}$-cell in the affine Weyl group. Let $\lambda \in {^L \mathfrak{h}}^* \cong \mathfrak{h}$ be an element of minimal length in such a region.

Canonical left $\tilde{S}$-region conjecture. The ideal $J(\lambda)$ is completely prime.

The conjecture proposes a source of completely prime ideals from Lusztig's $\tilde{S}$-regions. The paper notes that in types $B_2$ and $G_2$, the exceptional elements obtained from these regions are among those arising from Kazhdan–Lusztig cells; the general claim remains open.