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 $\chi \in {\mathfrak{g}}^{\ast }$ 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 Mœglin'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.

24 pages, minor changes, to appear in Compositio Mathematica