Minimal dimensional representations of reduced enveloping algebras for $mathfrak{gl}_n$

Let $mathfrak g = mathfrak{gl}_N(k)$, where $k$ is an algebraically closed field of characteristic $p > 0$, and $N in mathbb Z_{ge 1}$. Let $chi in mathfrak g^*$ and denote by $U_chi(mathfrak g)$ the corresponding reduced enveloping algebra. The Kac--Weisfeiler conjecture, which was proved by Premet, asserts that any finite dimensional $U_chi(mathfrak g)$-module has dimension divisible by $p^{d_chi}$, where $d_chi$ is half the dimension of the coadjoint orbit of $chi$. Our main theorem gives a classification of $U_chi(mathfrak g)$-modules of dimension $p^{d_chi}$. As a consequence, we deduce that they are all parabolically induced from a 1-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 M{oe}glins theorem on completely primitive ideals in $U(mathfrak{gl}_N(mathbb C))$. To obtain these results, we reduce to the case $chi$ is nilpotent, and then classify the 1-dimensional modules for the corresponding restricted $W$-algebra.