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 (AdG)-invariant symmetric bilinear form. Given a linear function χ on g we denote by U _χ(g) the reduced enveloping algebra of g associated with χ. By the Kac–Weisfeiler conjecture (now a theorem), any irreducible U _χ(g)-module has dimension divisible by p ^d(χ) where 2d(χ) is the dimension of the coadjoint G-orbit containing χ. 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 _χ(g) admits a module of dimension p ^d(χ).