Weak Premet–Topley conjecture on completely prime primitive ideals and minimal modules

Let ${\mathfrak g}_{\mathbb C}$ be a reductive Lie algebra over ${\mathbb C}$ with Chevalley basis ${\mathcal B}$, let $R={\mathbb Z}[1/q\mid q$ is a bad prime for $\Phi]$, and let ${\mathfrak g}_R$ and ${\mathfrak g}$ be the corresponding integral and characteristic-$p$ forms. Write ${\mathfrak t}_R$ for the induced Cartan lattice, $G_{\mathbb C}$ and $G$ for the corresponding reductive groups, and let $L_p^\chi(\lambda)$ denote the reduced Verma-module quotient defined in the setup. For a primitive ideal $I$ of $U({\mathfrak g}_{\mathbb C})$, call $I$ completely prime when $U({\mathfrak g}_{\mathbb C})/I$ is a domain, and write ${{\mathcal V}{\mathcal A}}'(I)$ for its associated variety. For each nilpotent coadjoint $G_{\mathbb C}$-orbit ${\mathbb O}_{\mathbb C}$ in ${\mathfrak g}_{\mathbb C}^{*}$, there exists a representative $\chi\in{\mathbb O}_{\mathbb C}$ mapping ${\mathfrak g}_R$ into $R$, and there exists $\widehat{\chi}\in G\cdot\overline{\chi}$, such that, if $L$ is a minimal-dimensional $U_{\widehat{\chi}}({\mathfrak g})$-module, then there exists $\lambda\in{\mathfrak t}_R^{*}$ satisfying

$$\operatorname{Ann}_{U({\mathfrak g}_{\mathbb C})}(L_{\mathbb C}(\lambda))\text{ is completely prime},$$ $${{\mathcal V}{\mathcal A}}'(\operatorname{Ann}_{U({\mathfrak g}_{\mathbb C})}(L_{\mathbb C}(\lambda)))=\overline{G_{\mathbb C}\cdot\chi},$$

and $L$ is a composition factor of $L_p^{\widehat{\chi}}(\lambda)$. This is presented as a weak version of the conjecture of Premet and Topley. The claim links minimal-dimensional modular representations to integral reductions of complex highest-weight modules with completely prime annihilators and prescribed associated varieties; its resolution status is not specified in the supplied material.