Minimal dimensional representations of reduced enveloping algebras for ${\mathfrak{g}\mathfrak{l}}_n$

Let $\mathfrak{g}={\mathfrak{g}\mathfrak{l}}_N(k)$, where $k$ is an algebraically closed field of characteristic $p>0$, and $N\in {\mathbb{Z}}_{\ge 1}$. Let $\chi \in \mathfrak g^*$anddenoteby$U_\chi(\mathfrak g)$ 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}glin’s theorem on completely primitive ideals in $U({\mathfrak{g}\mathfrak{l}}_N(\mathbb{C}))$. 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.