Modular representations of Lie algebras of reductive groups and Humphreys' conjecture

Let G be connected reductive algebraic group defined over an algebraically closed field of characteristic p > 0 and suppose that p is a good prime for the root system of G, the derived subgroup of G is simply connected and the Lie algebra g = Lie(G) admits a non-degenerate (Ad G)-invariant symmetric bilinear form. Given a linear function chi on g we denote by U-chi(g) the reduced enveloping algebra of g associated with chi. By the Kac-Weisfeiler conjecture (now a theorem), any irreducible U-chi(g)-module has dimension divisible by p(d(chi)) where 2d(chi) is the dimension of the coadjoint G-orbit containing chi. In this paper we give a positive answer to the natural question raised in the 1990s by Kac, Humphreys and the first-named author and show that any algebra U-chi(g) admits a module of dimension p(d(chi)). (C) 2021 Elsevier Inc. All rights reserved.