Minimal dimensional representations of reduced enveloping algebras for gl_n

Let g=gl_N(k), where k is an algebraically closed field of characteristic p>0, and N∈ℤ≥1. Let χ∈g^∗ and denote by Uχ(g) the corresponding reduced enveloping algebra. The Kac--Weisfeiler conjecture, which was proved by Premet, asserts that any finite dimensional U_χ(g)-module has dimension divisible by p^dχ, where d_χ is half the dimension of the coadjoint orbit of χ. Our main theorem gives a classification of U_χ(g)-modules of dimension p^dχ. As a consequence, we deduce that they are all parabolically induced from a 1-dimensional module for U₀(h) for a certain Levi subalgebra h of g; we view this as a modular analogue of Mœglin's theorem on completely primitive ideals in U(gl_N(ℂ)). To obtain these results, we reduce to the case χ is nilpotent, and then classify the 1-dimensional modules for the corresponding restricted W-algebra.