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.
Testing modules for irreducibility