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 Uχ(g) -module has dimension divisible by p\(^d\)χ , where dχ is half the dimension of the coadjoint orbit of χ . Our main theorem gives a classification of Uχ(g) -modules of dimension p\(^d\)χ. As a consequence, we deduce that they are all parabolically induced from a one-dimensional module for U\(_0\)(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 where χ is nilpotent, and then classify the one-dimensional modules for the corresponding restricted W -algebra.