Existence of integral one-dimensional modules for finite W-algebras

Let $\mathbb{O}$ be a special nilpotent orbit, and let $A$ and $\mathcal W$ denote the groups and finite W-algebra associated with the orbit as in the paper. The three exceptional orbits are $A_4+A_1$ in $E_7$, and $A_4+A_1$ and $E_6(a_1)+A_1$ in $E_8$, in Bala–Carter notation. Integral one-dimensional module conjecture. If $\mathbb{O}$ is not one of these three exceptional orbits, then there exists an $A$-stable one-dimensional $\mathcal W$-module with integral central character. For each of the three exceptional orbits, there exists a one-dimensional $\mathcal W$-module with integral central character. Premet's result gives $A$-stable one-dimensional representations in general, but not necessarily with integral central character; the exceptional cases are distinguished by the claimed failure of $A$-stable finite-dimensional representations with integral central character.