Let $\mathfrak{g}=\mathfrak{g}\mathfrak{l}_{N}(\Bbbk )$, where $\Bbbk$ is an algebraically closed field of characteristic $p>0$, and $N\in \mathbb{Z}_{{\geqslant}1}$. Let $\unicode[STIX]{x1D712}\in \mathfrak{g}^{\ast }$ and denote by $U_{\unicode[STIX]{x1D712}}(\mathfrak{g})$ the corresponding reduced enveloping algebra. The Kac–Weisfeiler conjecture, which was proved by Premet, asserts that any finite-dimensional $U_{\unicode[STIX]{x1D712}}(\mathfrak{g})$-module has dimension divisible by $p^{d_{\unicode[STIX]{x1D712}}}$, where $d_{\unicode[STIX]{x1D712}}$ is half the dimension of the coadjoint orbit of $\unicode[STIX]{x1D712}$. Our main theorem gives a classification of $U_{\unicode[STIX]{x1D712}}(\mathfrak{g})$-modules of dimension $p^{d_{\unicode[STIX]{x1D712}}}$. As a consequence, we deduce that they are all parabolically induced from a one-dimensional module for $U_{0}(\mathfrak{h})$ for a certain Levi subalgebra $\mathfrak{h}$ of $\mathfrak{g}$; we view this as a modular analogue of Mœglin’s theorem on completely primitive ideals in $U(\mathfrak{g}\mathfrak{l}_{N}(\mathbb{C}))$. To obtain these results, we reduce to the case where $\unicode[STIX]{x1D712}$ is nilpotent, and then classify the one-dimensional modules for the corresponding restricted $W$-algebra.

We use the theory of finite W-algebras associated to nilpotent orbits in the Lie algebra to give another proof of Mœglin’s theorem about completely prime primitive ideals in the enveloping algebra U(𝔤). We also make some new observations about Joseph’s Goldie rank polynomials in Cartan type A.