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

Abstract

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}glin's 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.