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 G, derived subgroup simply Lie algebra g=Lie(G) admits non-degenerate (AdG)-invariant symmetric bilinear form. Given linear function ? on g we denote by U?(g) reduced enveloping associated with ?. By Kac–Weisfeiler conjecture (now theorem), any irreducible U?(g)-module has dimension divisible pd(?) where 2d(?) coadjoint G-orbit containing In this paper give positive answer to natural question raised in 1990s Kac, Humphreys first-named author show module pd(?).